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Properties

Label 2-304-16.13-c1-0-13
Degree $2$
Conductor $304$
Sign $0.807 + 0.589i$
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 + 0.462i)2-s + (0.227 + 0.227i)3-s + (1.57 − 1.23i)4-s + (−2.67 + 2.67i)5-s + (−0.409 − 0.198i)6-s − 4.39i·7-s + (−1.52 + 2.37i)8-s − 2.89i·9-s + (2.33 − 4.81i)10-s + (2.02 − 2.02i)11-s + (0.639 + 0.0762i)12-s + (1.66 + 1.66i)13-s + (2.03 + 5.87i)14-s − 1.21·15-s + (0.941 − 3.88i)16-s + 4.30·17-s + ⋯
L(s)  = 1  + (−0.944 + 0.327i)2-s + (0.131 + 0.131i)3-s + (0.785 − 0.618i)4-s + (−1.19 + 1.19i)5-s + (−0.167 − 0.0811i)6-s − 1.66i·7-s + (−0.540 + 0.841i)8-s − 0.965i·9-s + (0.739 − 1.52i)10-s + (0.610 − 0.610i)11-s + (0.184 + 0.0220i)12-s + (0.461 + 0.461i)13-s + (0.543 + 1.57i)14-s − 0.314·15-s + (0.235 − 0.971i)16-s + 1.04·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.807 + 0.589i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.807 + 0.589i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.807 + 0.589i$
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: \((2,\ 304,\ (\ :1/2),\ 0.807 + 0.589i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.665098 - 0.216823i\)
\(L(\frac12)\) \(\approx\) \(0.665098 - 0.216823i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.33 - 0.462i)T \)
19 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (-0.227 - 0.227i)T + 3iT^{2} \)
5 \( 1 + (2.67 - 2.67i)T - 5iT^{2} \)
7 \( 1 + 4.39iT - 7T^{2} \)
11 \( 1 + (-2.02 + 2.02i)T - 11iT^{2} \)
13 \( 1 + (-1.66 - 1.66i)T + 13iT^{2} \)
17 \( 1 - 4.30T + 17T^{2} \)
23 \( 1 + 1.00iT - 23T^{2} \)
29 \( 1 + (3.73 + 3.73i)T + 29iT^{2} \)
31 \( 1 - 10.1T + 31T^{2} \)
37 \( 1 + (4.90 - 4.90i)T - 37iT^{2} \)
41 \( 1 + 9.29iT - 41T^{2} \)
43 \( 1 + (-5.74 + 5.74i)T - 43iT^{2} \)
47 \( 1 + 4.84T + 47T^{2} \)
53 \( 1 + (-3.75 + 3.75i)T - 53iT^{2} \)
59 \( 1 + (3.38 - 3.38i)T - 59iT^{2} \)
61 \( 1 + (7.04 + 7.04i)T + 61iT^{2} \)
67 \( 1 + (1.62 + 1.62i)T + 67iT^{2} \)
71 \( 1 - 2.97iT - 71T^{2} \)
73 \( 1 - 13.7iT - 73T^{2} \)
79 \( 1 + 5.25T + 79T^{2} \)
83 \( 1 + (-2.32 - 2.32i)T + 83iT^{2} \)
89 \( 1 + 3.66iT - 89T^{2} \)
97 \( 1 - 4.82T + 97T^{2} \)
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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

−11.40416318938332047420741414170, −10.56767170263278010651378926742, −9.929101749325527634956919162374, −8.636961416938732352152601746131, −7.69259016051801518415617348053, −6.94114745524518471787417575963, −6.30527898482893774677561890351, −4.03983052805044543208024341959, −3.29179899463893022226200500454, −0.75801192015733620283767210215, 1.52892259209321896057918812715, 3.10001205439589619975837483572, 4.64961491095366895901744542176, 5.89187049976341771393078381839, 7.53746292113813337118743654648, 8.215462401513083592304835083306, 8.809917464355305724597867514236, 9.683467038773592313309365834020, 11.01260148840497143469392344244, 11.99380789042129745677229772556

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