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Properties

Label 2-304-16.13-c1-0-18
Degree $2$
Conductor $304$
Sign $0.987 + 0.158i$
Analytic cond. $2.42745$
Root an. cond. $1.55802$
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  + (1.33 + 0.463i)2-s + (−1.27 − 1.27i)3-s + (1.56 + 1.23i)4-s + (1.80 − 1.80i)5-s + (−1.11 − 2.29i)6-s + 4.16i·7-s + (1.52 + 2.38i)8-s + 0.240i·9-s + (3.25 − 1.57i)10-s + (3.03 − 3.03i)11-s + (−0.420 − 3.57i)12-s + (−3.16 − 3.16i)13-s + (−1.93 + 5.56i)14-s − 4.60·15-s + (0.927 + 3.89i)16-s + 3.59·17-s + ⋯
L(s)  = 1  + (0.944 + 0.328i)2-s + (−0.734 − 0.734i)3-s + (0.784 + 0.619i)4-s + (0.809 − 0.809i)5-s + (−0.453 − 0.935i)6-s + 1.57i·7-s + (0.538 + 0.842i)8-s + 0.0803i·9-s + (1.02 − 0.499i)10-s + (0.914 − 0.914i)11-s + (−0.121 − 1.03i)12-s + (−0.876 − 0.876i)13-s + (−0.516 + 1.48i)14-s − 1.18·15-s + (0.231 + 0.972i)16-s + 0.871·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.158i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 + 0.158i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.987 + 0.158i$
Analytic conductor: \(2.42745\)
Root analytic conductor: \(1.55802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :1/2),\ 0.987 + 0.158i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.04934 - 0.162973i\)
\(L(\frac12)\) \(\approx\) \(2.04934 - 0.162973i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.33 - 0.463i)T \)
19 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (1.27 + 1.27i)T + 3iT^{2} \)
5 \( 1 + (-1.80 + 1.80i)T - 5iT^{2} \)
7 \( 1 - 4.16iT - 7T^{2} \)
11 \( 1 + (-3.03 + 3.03i)T - 11iT^{2} \)
13 \( 1 + (3.16 + 3.16i)T + 13iT^{2} \)
17 \( 1 - 3.59T + 17T^{2} \)
23 \( 1 - 1.75iT - 23T^{2} \)
29 \( 1 + (3.70 + 3.70i)T + 29iT^{2} \)
31 \( 1 + 6.05T + 31T^{2} \)
37 \( 1 + (0.874 - 0.874i)T - 37iT^{2} \)
41 \( 1 - 2.60iT - 41T^{2} \)
43 \( 1 + (6.74 - 6.74i)T - 43iT^{2} \)
47 \( 1 + 2.18T + 47T^{2} \)
53 \( 1 + (5.12 - 5.12i)T - 53iT^{2} \)
59 \( 1 + (-7.81 + 7.81i)T - 59iT^{2} \)
61 \( 1 + (-1.52 - 1.52i)T + 61iT^{2} \)
67 \( 1 + (-5.00 - 5.00i)T + 67iT^{2} \)
71 \( 1 - 1.97iT - 71T^{2} \)
73 \( 1 + 13.1iT - 73T^{2} \)
79 \( 1 + 13.6T + 79T^{2} \)
83 \( 1 + (10.5 + 10.5i)T + 83iT^{2} \)
89 \( 1 + 3.76iT - 89T^{2} \)
97 \( 1 - 16.8T + 97T^{2} \)
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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

−11.85466666775224288147566867080, −11.46708355523217108378795753177, −9.708598700104266394903108817960, −8.758316141088558586757367133815, −7.64168935348686783242185012274, −6.27696635170615984718785085218, −5.69654321759039304444935909753, −5.19216545101146741326593605135, −3.22856406035798539330130820886, −1.68154033023472108262627660964, 1.89377411840986075743985482221, 3.68914350471808050218108313575, 4.53627005474266020622791502804, 5.52838748708355822682154106908, 6.86868848174139691404086111930, 7.16599529558601148815177076790, 9.739737325540121666731747912419, 10.03472746581175363617802581999, 10.84132497491508288934307089098, 11.57403915768774428163867990746

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