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Properties

Label 2-308-44.43-c1-0-18
Degree $2$
Conductor $308$
Sign $0.954 - 0.297i$
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  + (0.963 − 1.03i)2-s + 3.04i·3-s + (−0.145 − 1.99i)4-s + 2.80·5-s + (3.15 + 2.93i)6-s + 7-s + (−2.20 − 1.77i)8-s − 6.29·9-s + (2.70 − 2.90i)10-s + (0.755 + 3.22i)11-s + (6.08 − 0.442i)12-s − 5.37i·13-s + (0.963 − 1.03i)14-s + 8.55i·15-s + (−3.95 + 0.578i)16-s + 5.95i·17-s + ⋯
L(s)  = 1  + (0.680 − 0.732i)2-s + 1.76i·3-s + (−0.0725 − 0.997i)4-s + 1.25·5-s + (1.28 + 1.19i)6-s + 0.377·7-s + (−0.779 − 0.626i)8-s − 2.09·9-s + (0.854 − 0.918i)10-s + (0.227 + 0.973i)11-s + (1.75 − 0.127i)12-s − 1.49i·13-s + (0.257 − 0.276i)14-s + 2.20i·15-s + (−0.989 + 0.144i)16-s + 1.44i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.297i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 - 0.297i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $0.954 - 0.297i$
Analytic conductor: \(2.45939\)
Root analytic conductor: \(1.56824\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{308} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 308,\ (\ :1/2),\ 0.954 - 0.297i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.08417 + 0.317665i\)
\(L(\frac12)\) \(\approx\) \(2.08417 + 0.317665i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.963 + 1.03i)T \)
7 \( 1 - T \)
11 \( 1 + (-0.755 - 3.22i)T \)
good3 \( 1 - 3.04iT - 3T^{2} \)
5 \( 1 - 2.80T + 5T^{2} \)
13 \( 1 + 5.37iT - 13T^{2} \)
17 \( 1 - 5.95iT - 17T^{2} \)
19 \( 1 + 0.325T + 19T^{2} \)
23 \( 1 + 1.56iT - 23T^{2} \)
29 \( 1 + 4.39iT - 29T^{2} \)
31 \( 1 + 3.00iT - 31T^{2} \)
37 \( 1 - 4.11T + 37T^{2} \)
41 \( 1 + 1.41iT - 41T^{2} \)
43 \( 1 + 12.4T + 43T^{2} \)
47 \( 1 + 9.32iT - 47T^{2} \)
53 \( 1 + 4.91T + 53T^{2} \)
59 \( 1 - 4.96iT - 59T^{2} \)
61 \( 1 + 10.7iT - 61T^{2} \)
67 \( 1 - 2.71iT - 67T^{2} \)
71 \( 1 + 1.79iT - 71T^{2} \)
73 \( 1 - 7.05iT - 73T^{2} \)
79 \( 1 + 1.63T + 79T^{2} \)
83 \( 1 + 5.50T + 83T^{2} \)
89 \( 1 - 5.20T + 89T^{2} \)
97 \( 1 - 16.1T + 97T^{2} \)
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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

−11.51585837304928823035451211085, −10.46008384927860398426874927852, −10.15591256628819277011359826699, −9.515558828995429433634891825030, −8.374183170864236601430425261747, −6.17890545247397969438388876168, −5.42431734298807919593719511037, −4.58523986663970937311307261881, −3.51599209506219783550498593221, −2.14619786898709527752320022483, 1.68147599820180683789020796450, 2.92427075267274348434556753758, 4.98711659462077989702910159210, 6.04131975082701357774111931637, 6.64214564959893221475654461748, 7.44897325795429686652267710160, 8.577007646200970247258632794949, 9.329250777860533703563289912222, 11.33841197973281248633552464724, 11.82351702081016615825272758926

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