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Properties

Label 2-308-28.19-c1-0-24
Degree $2$
Conductor $308$
Sign $0.983 - 0.179i$
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  + (1.18 − 0.777i)2-s + (1.22 + 2.12i)3-s + (0.791 − 1.83i)4-s + (−0.393 − 0.227i)5-s + (3.10 + 1.55i)6-s + (0.501 + 2.59i)7-s + (−0.492 − 2.78i)8-s + (−1.51 + 2.61i)9-s + (−0.641 + 0.0374i)10-s + (0.866 − 0.5i)11-s + (4.87 − 0.571i)12-s − 0.267i·13-s + (2.61 + 2.67i)14-s − 1.11i·15-s + (−2.74 − 2.90i)16-s + (−0.529 + 0.305i)17-s + ⋯
L(s)  = 1  + (0.835 − 0.549i)2-s + (0.708 + 1.22i)3-s + (0.395 − 0.918i)4-s + (−0.175 − 0.101i)5-s + (1.26 + 0.635i)6-s + (0.189 + 0.981i)7-s + (−0.174 − 0.984i)8-s + (−0.503 + 0.872i)9-s + (−0.202 + 0.0118i)10-s + (0.261 − 0.150i)11-s + (1.40 − 0.165i)12-s − 0.0743i·13-s + (0.698 + 0.716i)14-s − 0.287i·15-s + (−0.686 − 0.726i)16-s + (−0.128 + 0.0741i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.179i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 - 0.179i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $0.983 - 0.179i$
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: \((2,\ 308,\ (\ :1/2),\ 0.983 - 0.179i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.42426 + 0.219591i\)
\(L(\frac12)\) \(\approx\) \(2.42426 + 0.219591i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.18 + 0.777i)T \)
7 \( 1 + (-0.501 - 2.59i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
good3 \( 1 + (-1.22 - 2.12i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.393 + 0.227i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 + 0.267iT - 13T^{2} \)
17 \( 1 + (0.529 - 0.305i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.540 + 0.936i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.42 + 3.12i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 0.532T + 29T^{2} \)
31 \( 1 + (-3.41 - 5.91i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.63 - 6.30i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 11.8iT - 41T^{2} \)
43 \( 1 + 4.65iT - 43T^{2} \)
47 \( 1 + (1.34 - 2.33i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.44 + 4.23i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.96 - 8.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.09 + 4.09i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.17 + 0.680i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 0.0776iT - 71T^{2} \)
73 \( 1 + (-5.55 + 3.20i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-10.7 - 6.19i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 15.7T + 83T^{2} \)
89 \( 1 + (-8.78 - 5.07i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 10.9iT - 97T^{2} \)
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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

−11.89199768374034135840799087553, −10.70486126400695149766123976449, −10.01442746519771334292714875077, −9.075461366843716106834811611513, −8.295609272288418278794543378224, −6.51127536406617915323745982674, −5.33533651658033520893757026479, −4.41035440734726873142053033534, −3.43837103336228607929877320535, −2.29306370076575978458943081067, 1.82775102908752570410389314851, 3.32112240410393681509550299526, 4.41230168446580205898152305546, 5.98111765786584222151124660238, 6.96676848404340027921044724697, 7.66644382160913422991317303392, 8.216373601335922579206267123225, 9.642774910925267163872543573949, 11.15564264146504803867434530556, 11.93765512363402905968018215975

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