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Properties

Label 2-304-16.13-c1-0-30
Degree $2$
Conductor $304$
Sign $0.999 - 0.0314i$
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  + (1.35 + 0.399i)2-s + (0.0736 + 0.0736i)3-s + (1.68 + 1.08i)4-s + (1.57 − 1.57i)5-s + (0.0704 + 0.129i)6-s − 2.89i·7-s + (1.84 + 2.14i)8-s − 2.98i·9-s + (2.76 − 1.50i)10-s + (−4.00 + 4.00i)11-s + (0.0438 + 0.203i)12-s + (1.25 + 1.25i)13-s + (1.15 − 3.93i)14-s + 0.231·15-s + (1.64 + 3.64i)16-s − 4.21·17-s + ⋯
L(s)  = 1  + (0.959 + 0.282i)2-s + (0.0425 + 0.0425i)3-s + (0.840 + 0.542i)4-s + (0.704 − 0.704i)5-s + (0.0287 + 0.0527i)6-s − 1.09i·7-s + (0.652 + 0.757i)8-s − 0.996i·9-s + (0.874 − 0.476i)10-s + (−1.20 + 1.20i)11-s + (0.0126 + 0.0587i)12-s + (0.348 + 0.348i)13-s + (0.309 − 1.05i)14-s + 0.0598·15-s + (0.411 + 0.911i)16-s − 1.02·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0314i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0314i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.999 - 0.0314i$
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: \((2,\ 304,\ (\ :1/2),\ 0.999 - 0.0314i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.40907 + 0.0378995i\)
\(L(\frac12)\) \(\approx\) \(2.40907 + 0.0378995i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.35 - 0.399i)T \)
19 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (-0.0736 - 0.0736i)T + 3iT^{2} \)
5 \( 1 + (-1.57 + 1.57i)T - 5iT^{2} \)
7 \( 1 + 2.89iT - 7T^{2} \)
11 \( 1 + (4.00 - 4.00i)T - 11iT^{2} \)
13 \( 1 + (-1.25 - 1.25i)T + 13iT^{2} \)
17 \( 1 + 4.21T + 17T^{2} \)
23 \( 1 - 6.57iT - 23T^{2} \)
29 \( 1 + (-2.01 - 2.01i)T + 29iT^{2} \)
31 \( 1 + 3.23T + 31T^{2} \)
37 \( 1 + (2.15 - 2.15i)T - 37iT^{2} \)
41 \( 1 + 7.32iT - 41T^{2} \)
43 \( 1 + (-5.26 + 5.26i)T - 43iT^{2} \)
47 \( 1 + 7.26T + 47T^{2} \)
53 \( 1 + (-7.26 + 7.26i)T - 53iT^{2} \)
59 \( 1 + (0.840 - 0.840i)T - 59iT^{2} \)
61 \( 1 + (5.11 + 5.11i)T + 61iT^{2} \)
67 \( 1 + (8.49 + 8.49i)T + 67iT^{2} \)
71 \( 1 - 7.29iT - 71T^{2} \)
73 \( 1 + 0.267iT - 73T^{2} \)
79 \( 1 + 3.46T + 79T^{2} \)
83 \( 1 + (-10.6 - 10.6i)T + 83iT^{2} \)
89 \( 1 + 8.25iT - 89T^{2} \)
97 \( 1 - 14.4T + 97T^{2} \)
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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

−12.00876842526319885131733142909, −10.86867983200884762080120430022, −9.943466528484620445378693595696, −8.897222470113462253561962855042, −7.51357055782357743241148642638, −6.82657976258723397978564009341, −5.57595934299568111111521500787, −4.65076072898015865818267848252, −3.59833178145571519553525214086, −1.84592032031647372317582365444, 2.36244066615993582391036879395, 2.85814303159602257857459927512, 4.75640435060192757743860562584, 5.75072213967365469931433475730, 6.34823190632687931875898344954, 7.80084420440600932097897864422, 8.867849596129842920850434560435, 10.42371698638287889582636576788, 10.72793696976583292050030176317, 11.66946361816280181379803839680

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