Properties

Label 2-304-16.13-c1-0-21
Degree $2$
Conductor $304$
Sign $0.963 + 0.266i$
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  + (0.581 + 1.28i)2-s + (−0.951 − 0.951i)3-s + (−1.32 + 1.49i)4-s + (1.77 − 1.77i)5-s + (0.673 − 1.78i)6-s − 2.10i·7-s + (−2.70 − 0.836i)8-s − 1.18i·9-s + (3.32 + 1.25i)10-s + (1.29 − 1.29i)11-s + (2.68 − 0.166i)12-s + (1.24 + 1.24i)13-s + (2.72 − 1.22i)14-s − 3.38·15-s + (−0.492 − 3.96i)16-s + 3.50·17-s + ⋯
L(s)  = 1  + (0.411 + 0.911i)2-s + (−0.549 − 0.549i)3-s + (−0.662 + 0.749i)4-s + (0.795 − 0.795i)5-s + (0.275 − 0.726i)6-s − 0.797i·7-s + (−0.955 − 0.295i)8-s − 0.395i·9-s + (1.05 + 0.398i)10-s + (0.391 − 0.391i)11-s + (0.775 − 0.0479i)12-s + (0.344 + 0.344i)13-s + (0.727 − 0.327i)14-s − 0.874·15-s + (−0.123 − 0.992i)16-s + 0.849·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 + 0.266i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.963 + 0.266i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.963 + 0.266i$
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: \((2,\ 304,\ (\ :1/2),\ 0.963 + 0.266i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37223 - 0.185867i\)
\(L(\frac12)\) \(\approx\) \(1.37223 - 0.185867i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.581 - 1.28i)T \)
19 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (0.951 + 0.951i)T + 3iT^{2} \)
5 \( 1 + (-1.77 + 1.77i)T - 5iT^{2} \)
7 \( 1 + 2.10iT - 7T^{2} \)
11 \( 1 + (-1.29 + 1.29i)T - 11iT^{2} \)
13 \( 1 + (-1.24 - 1.24i)T + 13iT^{2} \)
17 \( 1 - 3.50T + 17T^{2} \)
23 \( 1 + 1.82iT - 23T^{2} \)
29 \( 1 + (-4.78 - 4.78i)T + 29iT^{2} \)
31 \( 1 + 2.23T + 31T^{2} \)
37 \( 1 + (-0.917 + 0.917i)T - 37iT^{2} \)
41 \( 1 + 6.01iT - 41T^{2} \)
43 \( 1 + (0.780 - 0.780i)T - 43iT^{2} \)
47 \( 1 + 3.26T + 47T^{2} \)
53 \( 1 + (7.27 - 7.27i)T - 53iT^{2} \)
59 \( 1 + (10.3 - 10.3i)T - 59iT^{2} \)
61 \( 1 + (5.66 + 5.66i)T + 61iT^{2} \)
67 \( 1 + (-6.96 - 6.96i)T + 67iT^{2} \)
71 \( 1 + 13.5iT - 71T^{2} \)
73 \( 1 - 0.342iT - 73T^{2} \)
79 \( 1 - 8.81T + 79T^{2} \)
83 \( 1 + (-9.85 - 9.85i)T + 83iT^{2} \)
89 \( 1 - 9.30iT - 89T^{2} \)
97 \( 1 - 1.31T + 97T^{2} \)
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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

−12.15790080885957350388520392639, −10.83283540061781353646670906049, −9.484505796613146387821874890139, −8.793079178405294214952110123922, −7.57176044566953520986774467967, −6.59166269343631143962480390434, −5.89233778100846266154374208307, −4.86303795665346365849992954264, −3.58836825647284218850589598596, −1.07499592752423604775978795854, 1.98496453110916363740399977084, 3.19465382929869432555158388944, 4.65622400831612562980543390590, 5.65713227362403598425384532583, 6.33726980384639355937899604468, 8.143312146938140302463274747563, 9.527332718553926206601717654668, 10.03454051344115334895592238797, 10.84770353265552618855968692504, 11.62831445520423513112184951069

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