Properties

Label 1-680-680.517-r0-0-0
Degree 11
Conductor 680680
Sign 0.6570.753i0.657 - 0.753i
Analytic cond. 3.157903.15790
Root an. cond. 3.157903.15790
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.923 − 0.382i)3-s + (0.382 − 0.923i)7-s + (0.707 − 0.707i)9-s + (−0.382 + 0.923i)11-s + 13-s + (−0.707 − 0.707i)19-s i·21-s + (0.923 + 0.382i)23-s + (0.382 − 0.923i)27-s + (0.923 − 0.382i)29-s + (0.382 + 0.923i)31-s + i·33-s + (−0.923 + 0.382i)37-s + (0.923 − 0.382i)39-s + (−0.923 − 0.382i)41-s + ⋯
L(s)  = 1  + (0.923 − 0.382i)3-s + (0.382 − 0.923i)7-s + (0.707 − 0.707i)9-s + (−0.382 + 0.923i)11-s + 13-s + (−0.707 − 0.707i)19-s i·21-s + (0.923 + 0.382i)23-s + (0.382 − 0.923i)27-s + (0.923 − 0.382i)29-s + (0.382 + 0.923i)31-s + i·33-s + (−0.923 + 0.382i)37-s + (0.923 − 0.382i)39-s + (−0.923 − 0.382i)41-s + ⋯

Functional equation

Λ(s)=(680s/2ΓR(s)L(s)=((0.6570.753i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.657 - 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(680s/2ΓR(s)L(s)=((0.6570.753i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.657 - 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 680680    =    235172^{3} \cdot 5 \cdot 17
Sign: 0.6570.753i0.657 - 0.753i
Analytic conductor: 3.157903.15790
Root analytic conductor: 3.157903.15790
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ680(517,)\chi_{680} (517, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 680, (0: ), 0.6570.753i)(1,\ 680,\ (0:\ ),\ 0.657 - 0.753i)

Particular Values

L(12)L(\frac{1}{2}) \approx 1.9517459240.8864848242i1.951745924 - 0.8864848242i
L(12)L(\frac12) \approx 1.9517459240.8864848242i1.951745924 - 0.8864848242i
L(1)L(1) \approx 1.4982357180.3593236479i1.498235718 - 0.3593236479i
L(1)L(1) \approx 1.4982357180.3593236479i1.498235718 - 0.3593236479i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
5 1 1
17 1 1
good3 1+(0.9230.382i)T 1 + (0.923 - 0.382i)T
7 1+(0.3820.923i)T 1 + (0.382 - 0.923i)T
11 1+(0.382+0.923i)T 1 + (-0.382 + 0.923i)T
13 1+T 1 + T
19 1+(0.7070.707i)T 1 + (-0.707 - 0.707i)T
23 1+(0.923+0.382i)T 1 + (0.923 + 0.382i)T
29 1+(0.9230.382i)T 1 + (0.923 - 0.382i)T
31 1+(0.382+0.923i)T 1 + (0.382 + 0.923i)T
37 1+(0.923+0.382i)T 1 + (-0.923 + 0.382i)T
41 1+(0.9230.382i)T 1 + (-0.923 - 0.382i)T
43 1+(0.7070.707i)T 1 + (-0.707 - 0.707i)T
47 1+T 1 + T
53 1+(0.7070.707i)T 1 + (0.707 - 0.707i)T
59 1+(0.7070.707i)T 1 + (0.707 - 0.707i)T
61 1+(0.9230.382i)T 1 + (-0.923 - 0.382i)T
67 1iT 1 - iT
71 1+(0.3820.923i)T 1 + (-0.382 - 0.923i)T
73 1+(0.382+0.923i)T 1 + (0.382 + 0.923i)T
79 1+(0.382+0.923i)T 1 + (-0.382 + 0.923i)T
83 1+(0.707+0.707i)T 1 + (-0.707 + 0.707i)T
89 1iT 1 - iT
97 1+(0.382+0.923i)T 1 + (0.382 + 0.923i)T
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   L(s)=p (1αpps)1L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−22.770321424930024801300846152167, −21.68492671126937770332101346703, −21.19048955434083578445313873025, −20.63629497396399881852087479756, −19.505403526336452651538410428111, −18.72548023839555976920011503088, −18.32916683937249851731173831920, −16.92684994438717084128747288591, −16.06176205203236332587900222368, −15.35837627772345928994639399801, −14.660219850226667699098023845871, −13.74674426833237312651837676137, −13.06353876700088067326685890187, −11.98560837513914083331029675154, −10.92357833573515209639195777648, −10.25202023207020860647481967577, −8.91068083464219236701274220265, −8.613237750456424285855139771493, −7.79863829012998536068595057763, −6.42643610022688948841898380715, −5.484914201153367834124953849527, −4.44361347217686753231951839811, −3.35875915754877970916389640691, −2.574769907689064180883670249695, −1.43424075172300046559203932159, 1.07220258290206145248102867384, 2.04791405354760605270277428563, 3.213959446321922453163615488349, 4.14052023482100091183224689898, 5.058001619288524828857361239997, 6.71431740028138240021190394841, 7.112058381196982128101130776973, 8.23409812608486022980792998080, 8.80675958108173701169453582766, 10.01761218283718099739918216501, 10.66366989510964171981645468550, 11.83383190049601272896880030380, 12.88626500680980720628977264642, 13.52365363212984159930960599570, 14.17961758269109634602043933468, 15.227795115952633790475808879975, 15.711870494419968076267772513354, 17.106372016552229481276850150495, 17.72008266387827692615502131445, 18.60603400009921021416773973977, 19.4705519671999203450648122845, 20.20983564496752834481033332973, 20.85663687909638034435219035888, 21.465263654830789679100661040619, 22.93130952895219350388712913967

Graph of the ZZ-function along the critical line