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Properties

Label 2-308-28.19-c1-0-23
Degree $2$
Conductor $308$
Sign $0.783 - 0.621i$
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  + (1.20 + 0.741i)2-s + (−0.246 − 0.427i)3-s + (0.899 + 1.78i)4-s + (1.30 + 0.752i)5-s + (0.0199 − 0.697i)6-s + (0.506 − 2.59i)7-s + (−0.242 + 2.81i)8-s + (1.37 − 2.38i)9-s + (1.01 + 1.87i)10-s + (0.866 − 0.5i)11-s + (0.541 − 0.824i)12-s + 5.78i·13-s + (2.53 − 2.75i)14-s − 0.742i·15-s + (−2.38 + 3.21i)16-s + (−2.85 + 1.64i)17-s + ⋯
L(s)  = 1  + (0.851 + 0.524i)2-s + (−0.142 − 0.246i)3-s + (0.449 + 0.893i)4-s + (0.582 + 0.336i)5-s + (0.00815 − 0.284i)6-s + (0.191 − 0.981i)7-s + (−0.0858 + 0.996i)8-s + (0.459 − 0.795i)9-s + (0.319 + 0.592i)10-s + (0.261 − 0.150i)11-s + (0.156 − 0.238i)12-s + 1.60i·13-s + (0.677 − 0.735i)14-s − 0.191i·15-s + (−0.595 + 0.803i)16-s + (−0.691 + 0.399i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.783 - 0.621i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.783 - 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $0.783 - 0.621i$
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: \((2,\ 308,\ (\ :1/2),\ 0.783 - 0.621i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.07092 + 0.721388i\)
\(L(\frac12)\) \(\approx\) \(2.07092 + 0.721388i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.20 - 0.741i)T \)
7 \( 1 + (-0.506 + 2.59i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
good3 \( 1 + (0.246 + 0.427i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.30 - 0.752i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 - 5.78iT - 13T^{2} \)
17 \( 1 + (2.85 - 1.64i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.365 - 0.633i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.37 + 1.94i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 6.68T + 29T^{2} \)
31 \( 1 + (2.79 + 4.83i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.511 + 0.885i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 0.0235iT - 41T^{2} \)
43 \( 1 + 6.96iT - 43T^{2} \)
47 \( 1 + (3.32 - 5.75i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.59 + 2.76i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.557 + 0.966i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (9.43 + 5.44i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.50 - 4.91i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.98iT - 71T^{2} \)
73 \( 1 + (-4.41 + 2.54i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-13.2 - 7.64i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.48T + 83T^{2} \)
89 \( 1 + (4.54 + 2.62i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 18.5iT - 97T^{2} \)
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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

−11.94249976039877956305235279421, −11.11241905942354055273407485987, −9.998355911531454344206451130054, −8.860511759425836120962393728478, −7.59688000697006103133825335916, −6.54222691759390794229525219120, −6.30432517512824291004864366840, −4.52728081710539418279271125372, −3.82505189631536823647151666471, −1.99093735174331557016763557336, 1.78804926353580993744687549895, 3.03041532252878557263198858560, 4.68973910714010072376432721454, 5.34791054578578066055291697472, 6.24779198749123192436435422293, 7.72408756765613832108915322334, 9.028089521051984177861271532151, 9.995200179034162433914688145613, 10.71485125176146004212598027856, 11.70996730212138572449046558944

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