Properties

Label 2-304-16.13-c1-0-34
Degree $2$
Conductor $304$
Sign $-0.868 + 0.495i$
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  + (1.15 − 0.822i)2-s + (−1.99 − 1.99i)3-s + (0.647 − 1.89i)4-s + (2.01 − 2.01i)5-s + (−3.94 − 0.656i)6-s + 0.456i·7-s + (−0.810 − 2.70i)8-s + 4.98i·9-s + (0.662 − 3.97i)10-s + (−0.0937 + 0.0937i)11-s + (−5.07 + 2.48i)12-s + (3.81 + 3.81i)13-s + (0.375 + 0.525i)14-s − 8.06·15-s + (−3.16 − 2.45i)16-s − 5.00·17-s + ⋯
L(s)  = 1  + (0.813 − 0.581i)2-s + (−1.15 − 1.15i)3-s + (0.323 − 0.946i)4-s + (0.901 − 0.901i)5-s + (−1.60 − 0.267i)6-s + 0.172i·7-s + (−0.286 − 0.958i)8-s + 1.66i·9-s + (0.209 − 1.25i)10-s + (−0.0282 + 0.0282i)11-s + (−1.46 + 0.717i)12-s + (1.05 + 1.05i)13-s + (0.100 + 0.140i)14-s − 2.08·15-s + (−0.790 − 0.612i)16-s − 1.21·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.868 + 0.495i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.868 + 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.868 + 0.495i$
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: \((2,\ 304,\ (\ :1/2),\ -0.868 + 0.495i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.405491 - 1.52946i\)
\(L(\frac12)\) \(\approx\) \(0.405491 - 1.52946i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.15 + 0.822i)T \)
19 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (1.99 + 1.99i)T + 3iT^{2} \)
5 \( 1 + (-2.01 + 2.01i)T - 5iT^{2} \)
7 \( 1 - 0.456iT - 7T^{2} \)
11 \( 1 + (0.0937 - 0.0937i)T - 11iT^{2} \)
13 \( 1 + (-3.81 - 3.81i)T + 13iT^{2} \)
17 \( 1 + 5.00T + 17T^{2} \)
23 \( 1 + 3.03iT - 23T^{2} \)
29 \( 1 + (-4.15 - 4.15i)T + 29iT^{2} \)
31 \( 1 - 3.17T + 31T^{2} \)
37 \( 1 + (-8.33 + 8.33i)T - 37iT^{2} \)
41 \( 1 + 0.444iT - 41T^{2} \)
43 \( 1 + (3.04 - 3.04i)T - 43iT^{2} \)
47 \( 1 + 9.38T + 47T^{2} \)
53 \( 1 + (0.997 - 0.997i)T - 53iT^{2} \)
59 \( 1 + (-6.43 + 6.43i)T - 59iT^{2} \)
61 \( 1 + (7.30 + 7.30i)T + 61iT^{2} \)
67 \( 1 + (-8.26 - 8.26i)T + 67iT^{2} \)
71 \( 1 - 12.3iT - 71T^{2} \)
73 \( 1 - 4.07iT - 73T^{2} \)
79 \( 1 - 9.04T + 79T^{2} \)
83 \( 1 + (6.04 + 6.04i)T + 83iT^{2} \)
89 \( 1 + 9.07iT - 89T^{2} \)
97 \( 1 + 17.0T + 97T^{2} \)
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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

−11.46172904871914467079920395747, −10.93095917849029626257402005839, −9.624150872723930185503969730342, −8.587964432542464310355206942037, −6.79965000141514407659399878840, −6.25282567792942246206264002942, −5.38382682190588555231973737017, −4.40250127932279973592789540904, −2.15566668041700517628672737426, −1.14373228230131347010467088424, 2.92148816994210938647449851224, 4.16629183996425943635897077861, 5.22216164622152665134131703168, 6.13814265610775931664717012765, 6.59645439919082913751782693513, 8.186773577036853678978473240619, 9.582432085534346152523895481581, 10.52162580969463540798352250481, 11.09172639557032617704156208252, 11.91177009285324990370343248393

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