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Properties

Label 2-296-296.85-c1-0-30
Degree $2$
Conductor $296$
Sign $0.896 + 0.443i$
Analytic cond. $2.36357$
Root an. cond. $1.53739$
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 − 0.392i)2-s + (0.727 + 0.420i)3-s + (1.69 − 1.06i)4-s + (0.249 − 0.432i)5-s + (1.15 + 0.285i)6-s + (0.235 − 0.407i)7-s + (1.88 − 2.11i)8-s + (−1.14 − 1.98i)9-s + (0.169 − 0.685i)10-s + 1.14i·11-s + (1.67 − 0.0646i)12-s + (−3.10 + 5.37i)13-s + (0.159 − 0.646i)14-s + (0.363 − 0.209i)15-s + (1.72 − 3.60i)16-s + (−0.363 + 0.209i)17-s + ⋯
L(s)  = 1  + (0.960 − 0.277i)2-s + (0.420 + 0.242i)3-s + (0.846 − 0.532i)4-s + (0.111 − 0.193i)5-s + (0.470 + 0.116i)6-s + (0.0889 − 0.154i)7-s + (0.665 − 0.746i)8-s + (−0.382 − 0.662i)9-s + (0.0536 − 0.216i)10-s + 0.346i·11-s + (0.484 − 0.0186i)12-s + (−0.861 + 1.49i)13-s + (0.0427 − 0.172i)14-s + (0.0937 − 0.0541i)15-s + (0.431 − 0.901i)16-s + (−0.0881 + 0.0508i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.896 + 0.443i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.896 + 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(296\)    =    \(2^{3} \cdot 37\)
Sign: $0.896 + 0.443i$
Analytic conductor: \(2.36357\)
Root analytic conductor: \(1.53739\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{296} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 296,\ (\ :1/2),\ 0.896 + 0.443i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.41658 - 0.565431i\)
\(L(\frac12)\) \(\approx\) \(2.41658 - 0.565431i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.35 + 0.392i)T \)
37 \( 1 + (5.34 - 2.90i)T \)
good3 \( 1 + (-0.727 - 0.420i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.249 + 0.432i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.235 + 0.407i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - 1.14iT - 11T^{2} \)
13 \( 1 + (3.10 - 5.37i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.363 - 0.209i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.39 + 2.41i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 7.74iT - 23T^{2} \)
29 \( 1 + 5.48T + 29T^{2} \)
31 \( 1 + 7.40iT - 31T^{2} \)
41 \( 1 + (-1.51 + 2.61i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 1.44T + 43T^{2} \)
47 \( 1 + 0.412T + 47T^{2} \)
53 \( 1 + (6.89 - 3.98i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.73 + 9.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.66 + 4.62i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.66 - 5.00i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.85 + 4.94i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 5.09T + 73T^{2} \)
79 \( 1 + (-5.88 - 3.39i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-11.4 + 6.63i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (10.0 - 5.80i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 8.36iT - 97T^{2} \)
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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

−11.71210116822765485703427522306, −11.13536081922160915747505451917, −9.571343892316556800516952036506, −9.338990995007024812066100002098, −7.57819909843212481738035429198, −6.70543104862531724108505236390, −5.46101142415887051611928176693, −4.39163985640075956692283161225, −3.35598851081208381578713843260, −1.91771258799043687497159019276, 2.35262631940459937586014086656, 3.26306550065223765541691652716, 4.88296392686743763682305005574, 5.68423050371746997785378485793, 6.91723355100902994522574777407, 7.904473999726819165837067143891, 8.577560006971596775091384494378, 10.30378912350638515725274951409, 10.91376901980250347641784266266, 12.21160550329829020031426634136

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