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Properties

Label 2-296-296.101-c1-0-31
Degree $2$
Conductor $296$
Sign $0.956 + 0.290i$
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  + (1.26 + 0.632i)2-s + (1.98 − 1.14i)3-s + (1.19 + 1.60i)4-s + (−1.77 − 3.07i)5-s + (3.22 − 0.193i)6-s + (−0.968 − 1.67i)7-s + (0.505 + 2.78i)8-s + (1.11 − 1.93i)9-s + (−0.300 − 5.01i)10-s + 1.93i·11-s + (4.20 + 1.79i)12-s + (1.72 + 2.99i)13-s + (−0.163 − 2.73i)14-s + (−7.03 − 4.06i)15-s + (−1.12 + 3.83i)16-s + (2.13 + 1.23i)17-s + ⋯
L(s)  = 1  + (0.894 + 0.447i)2-s + (1.14 − 0.660i)3-s + (0.599 + 0.800i)4-s + (−0.793 − 1.37i)5-s + (1.31 − 0.0789i)6-s + (−0.366 − 0.634i)7-s + (0.178 + 0.983i)8-s + (0.372 − 0.644i)9-s + (−0.0949 − 1.58i)10-s + 0.582i·11-s + (1.21 + 0.519i)12-s + (0.479 + 0.829i)13-s + (−0.0437 − 0.730i)14-s + (−1.81 − 1.04i)15-s + (−0.280 + 0.959i)16-s + (0.518 + 0.299i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.290i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.956 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(296\)    =    \(2^{3} \cdot 37\)
Sign: $0.956 + 0.290i$
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: \((2,\ 296,\ (\ :1/2),\ 0.956 + 0.290i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.52911 - 0.375289i\)
\(L(\frac12)\) \(\approx\) \(2.52911 - 0.375289i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.26 - 0.632i)T \)
37 \( 1 + (4.63 + 3.93i)T \)
good3 \( 1 + (-1.98 + 1.14i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.77 + 3.07i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.968 + 1.67i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 1.93iT - 11T^{2} \)
13 \( 1 + (-1.72 - 2.99i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.13 - 1.23i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.29 + 2.23i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 8.62iT - 23T^{2} \)
29 \( 1 + 5.38T + 29T^{2} \)
31 \( 1 + 8.79iT - 31T^{2} \)
41 \( 1 + (1.00 + 1.74i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 3.46T + 43T^{2} \)
47 \( 1 - 10.3T + 47T^{2} \)
53 \( 1 + (1.03 + 0.596i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.46 - 2.53i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.10 + 8.83i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.08 + 0.627i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.62 - 2.81i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 8.44T + 73T^{2} \)
79 \( 1 + (-1.69 + 0.975i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.68 + 4.43i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (9.55 + 5.51i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 17.3iT - 97T^{2} \)
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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

−12.12669565051126522634208736258, −11.21602171054636315943266383372, −9.378152022267214593847862982500, −8.627766171640582999850803037927, −7.60655690231647180714294272033, −7.24520803369230498655141117664, −5.63325438007162056275532929970, −4.26897680496037958451044351660, −3.60028034033986544552140916492, −1.81421502773226378388818205072, 2.74864851365964799596275976107, 3.18926098672262516462919223968, 4.11169592655756724272826959370, 5.74537439486152323909685085234, 6.80216651975980074991595043537, 8.027490229609363528739682945891, 9.037223866320260194856903771929, 10.39331976727896369425695330162, 10.64306200739542044815291048369, 11.86411347534785869599941560523

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