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Properties

Label 1-680-680.517-r0-0-0
Degree $1$
Conductor $680$
Sign $0.657 - 0.753i$
Analytic cond. $3.15790$
Root an. cond. $3.15790$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(1\)
Conductor: \(680\)    =    \(2^{3} \cdot 5 \cdot 17\)
Sign: $0.657 - 0.753i$
Analytic conductor: \(3.15790\)
Root analytic conductor: \(3.15790\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{680} (517, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 680,\ (0:\ ),\ 0.657 - 0.753i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.951745924 - 0.8864848242i\)
\(L(\frac12)\) \(\approx\) \(1.951745924 - 0.8864848242i\)
\(L(1)\) \(\approx\) \(1.498235718 - 0.3593236479i\)
\(L(1)\) \(\approx\) \(1.498235718 - 0.3593236479i\)

Euler product

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

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 $Z$-function along the critical line