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Properties

Label 2-308-77.41-c1-0-6
Degree 22
Conductor 308308
Sign 0.400+0.916i0.400 + 0.916i
Analytic cond. 2.459392.45939
Root an. cond. 1.568241.56824
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.359 + 0.116i)3-s + (1.61 − 2.22i)5-s + (0.904 − 2.48i)7-s + (−2.31 + 1.67i)9-s + (−2.11 − 2.55i)11-s + (1.07 − 0.778i)13-s + (−0.321 + 0.989i)15-s + (2.01 + 1.46i)17-s + (−1.78 − 5.48i)19-s + (−0.0347 + 0.999i)21-s + 5.62·23-s + (−0.795 − 2.44i)25-s + (1.30 − 1.79i)27-s + (6.07 + 1.97i)29-s + (−0.830 − 1.14i)31-s + ⋯
L(s)  = 1  + (−0.207 + 0.0674i)3-s + (0.723 − 0.995i)5-s + (0.341 − 0.939i)7-s + (−0.770 + 0.559i)9-s + (−0.638 − 0.769i)11-s + (0.297 − 0.215i)13-s + (−0.0829 + 0.255i)15-s + (0.488 + 0.354i)17-s + (−0.409 − 1.25i)19-s + (−0.00758 + 0.218i)21-s + 1.17·23-s + (−0.159 − 0.489i)25-s + (0.250 − 0.344i)27-s + (1.12 + 0.366i)29-s + (−0.149 − 0.205i)31-s + ⋯

Functional equation

Λ(s)=(308s/2 ΓC(s) L(s)=((0.400+0.916i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.400 + 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(308s/2 ΓC(s+1/2) L(s)=((0.400+0.916i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.400 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 308308    =    22⋅7⋅112^{2} \cdot 7 \cdot 11
Sign: 0.400+0.916i0.400 + 0.916i
Analytic conductor: 2.459392.45939
Root analytic conductor: 1.568241.56824
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ308(41,⋅)\chi_{308} (41, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 308, ( :1/2), 0.400+0.916i)(2,\ 308,\ (\ :1/2),\ 0.400 + 0.916i)

Particular Values

L(1)L(1) ≈\approx 1.07430−0.702479i1.07430 - 0.702479i
L(12)L(\frac12) ≈\approx 1.07430−0.702479i1.07430 - 0.702479i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
7 1+(−0.904+2.48i)T 1 + (-0.904 + 2.48i)T
11 1+(2.11+2.55i)T 1 + (2.11 + 2.55i)T
good3 1+(0.359−0.116i)T+(2.42−1.76i)T2 1 + (0.359 - 0.116i)T + (2.42 - 1.76i)T^{2}
5 1+(−1.61+2.22i)T+(−1.54−4.75i)T2 1 + (-1.61 + 2.22i)T + (-1.54 - 4.75i)T^{2}
13 1+(−1.07+0.778i)T+(4.01−12.3i)T2 1 + (-1.07 + 0.778i)T + (4.01 - 12.3i)T^{2}
17 1+(−2.01−1.46i)T+(5.25+16.1i)T2 1 + (-2.01 - 1.46i)T + (5.25 + 16.1i)T^{2}
19 1+(1.78+5.48i)T+(−15.3+11.1i)T2 1 + (1.78 + 5.48i)T + (-15.3 + 11.1i)T^{2}
23 1−5.62T+23T2 1 - 5.62T + 23T^{2}
29 1+(−6.07−1.97i)T+(23.4+17.0i)T2 1 + (-6.07 - 1.97i)T + (23.4 + 17.0i)T^{2}
31 1+(0.830+1.14i)T+(−9.57+29.4i)T2 1 + (0.830 + 1.14i)T + (-9.57 + 29.4i)T^{2}
37 1+(−0.224+0.690i)T+(−29.9−21.7i)T2 1 + (-0.224 + 0.690i)T + (-29.9 - 21.7i)T^{2}
41 1+(−1.87−5.78i)T+(−33.1+24.0i)T2 1 + (-1.87 - 5.78i)T + (-33.1 + 24.0i)T^{2}
43 1−8.58iT−43T2 1 - 8.58iT - 43T^{2}
47 1+(6.76−2.19i)T+(38.0−27.6i)T2 1 + (6.76 - 2.19i)T + (38.0 - 27.6i)T^{2}
53 1+(3.60−2.61i)T+(16.3−50.4i)T2 1 + (3.60 - 2.61i)T + (16.3 - 50.4i)T^{2}
59 1+(6.72+2.18i)T+(47.7+34.6i)T2 1 + (6.72 + 2.18i)T + (47.7 + 34.6i)T^{2}
61 1+(−7.87−5.72i)T+(18.8+58.0i)T2 1 + (-7.87 - 5.72i)T + (18.8 + 58.0i)T^{2}
67 1+2.85T+67T2 1 + 2.85T + 67T^{2}
71 1+(−6.22−4.52i)T+(21.9+67.5i)T2 1 + (-6.22 - 4.52i)T + (21.9 + 67.5i)T^{2}
73 1+(−1.51+4.67i)T+(−59.0−42.9i)T2 1 + (-1.51 + 4.67i)T + (-59.0 - 42.9i)T^{2}
79 1+(−5.03−6.93i)T+(−24.4+75.1i)T2 1 + (-5.03 - 6.93i)T + (-24.4 + 75.1i)T^{2}
83 1+(−9.86−7.16i)T+(25.6+78.9i)T2 1 + (-9.86 - 7.16i)T + (25.6 + 78.9i)T^{2}
89 1−16.4iT−89T2 1 - 16.4iT - 89T^{2}
97 1+(6.62+9.12i)T+(−29.9+92.2i)T2 1 + (6.62 + 9.12i)T + (-29.9 + 92.2i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−11.16707070250958305465498824383, −10.84714193366158875550837957925, −9.652168687835728542100056350112, −8.597066280691429125447755420248, −7.938006145950144096881164575549, −6.48391191360803653372318011950, −5.33903040386580139688862578656, −4.69124331116530929821270890220, −2.91812758209204116462006671122, −1.03347411044576443169492305539, 2.13222321754517821113748917503, 3.21609429145581746316007744316, 5.09174540226769492549504721056, 5.98634026434679871041306636150, 6.81249732776702675227047945059, 8.100258076265280961011100976970, 9.116238331509453795579038462648, 10.10125417427440328655731970534, 10.88629953026208793473996821798, 11.91319039558764072288035629355

Graph of the ZZ-function along the critical line