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Properties

Label 2-308-28.19-c1-0-13
Degree $2$
Conductor $308$
Sign $0.690 + 0.723i$
Analytic cond. $2.45939$
Root an. cond. $1.56824$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.36 − 0.368i)2-s + (0.591 + 1.02i)3-s + (1.72 + 1.00i)4-s + (−3.61 − 2.08i)5-s + (−0.430 − 1.61i)6-s + (2.05 + 1.66i)7-s + (−1.99 − 2.00i)8-s + (0.800 − 1.38i)9-s + (4.16 + 4.17i)10-s + (0.866 − 0.5i)11-s + (−0.00730 + 2.36i)12-s − 3.06i·13-s + (−2.19 − 3.02i)14-s − 4.93i·15-s + (1.97 + 3.47i)16-s + (3.05 − 1.76i)17-s + ⋯
L(s)  = 1  + (−0.965 − 0.260i)2-s + (0.341 + 0.591i)3-s + (0.864 + 0.502i)4-s + (−1.61 − 0.932i)5-s + (−0.175 − 0.660i)6-s + (0.778 + 0.628i)7-s + (−0.703 − 0.710i)8-s + (0.266 − 0.461i)9-s + (1.31 + 1.32i)10-s + (0.261 − 0.150i)11-s + (−0.00210 + 0.683i)12-s − 0.849i·13-s + (−0.587 − 0.809i)14-s − 1.27i·15-s + (0.494 + 0.869i)16-s + (0.742 − 0.428i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.690 + 0.723i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.690 + 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $0.690 + 0.723i$
Analytic conductor: \(2.45939\)
Root analytic conductor: \(1.56824\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{308} (243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 308,\ (\ :1/2),\ 0.690 + 0.723i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.737133 - 0.315579i\)
\(L(\frac12)\) \(\approx\) \(0.737133 - 0.315579i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 + 0.368i)T \)
7 \( 1 + (-2.05 - 1.66i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
good3 \( 1 + (-0.591 - 1.02i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (3.61 + 2.08i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 + 3.06iT - 13T^{2} \)
17 \( 1 + (-3.05 + 1.76i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.82 + 3.16i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.09 - 2.36i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.48T + 29T^{2} \)
31 \( 1 + (2.78 + 4.82i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.17 + 7.23i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 6.95iT - 41T^{2} \)
43 \( 1 + 2.12iT - 43T^{2} \)
47 \( 1 + (2.26 - 3.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.58 - 4.47i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.43 - 2.48i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (11.0 + 6.38i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (12.9 - 7.46i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.35iT - 71T^{2} \)
73 \( 1 + (-5.79 + 3.34i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.60 + 3.23i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.95T + 83T^{2} \)
89 \( 1 + (-0.589 - 0.340i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 11.5iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.53248011747734830876049313767, −10.74011864791815437134735899871, −9.229776093877317810151632709589, −9.003475990119469563891899168855, −7.88812776100396303856165510945, −7.37691716929255185223516099316, −5.41973733809083547203728120451, −4.13931224738479217010086320632, −3.12251981667400155102088704689, −0.890623650280212272621335348360, 1.49061646881028449669029742359, 3.20307544405373374249106314587, 4.61657143585926673628502394888, 6.58749488159568924249908721420, 7.34170239938887726746159755415, 7.83500310189937795442058622111, 8.562436997070348724000877547478, 10.09431985475979635682835912297, 10.90650399625631218784193985406, 11.55278845577409919830131045468

Graph of the $Z$-function along the critical line