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Properties

Label 2-296-296.101-c1-0-20
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: $\chi_{296} (101, \cdot )$
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

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

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