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Properties

Label 2-296-296.101-c1-0-18
Degree $2$
Conductor $296$
Sign $-0.0965 + 0.995i$
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  + (−1.38 + 0.307i)2-s + (−1.31 + 0.759i)3-s + (1.81 − 0.849i)4-s + (0.762 + 1.31i)5-s + (1.58 − 1.45i)6-s + (−2.35 − 4.08i)7-s + (−2.23 + 1.73i)8-s + (−0.344 + 0.597i)9-s + (−1.45 − 1.58i)10-s + 0.210i·11-s + (−1.73 + 2.49i)12-s + (−0.457 − 0.792i)13-s + (4.51 + 4.91i)14-s + (−2.00 − 1.15i)15-s + (2.55 − 3.07i)16-s + (−2.84 − 1.64i)17-s + ⋯
L(s)  = 1  + (−0.976 + 0.217i)2-s + (−0.759 + 0.438i)3-s + (0.905 − 0.424i)4-s + (0.340 + 0.590i)5-s + (0.646 − 0.593i)6-s + (−0.891 − 1.54i)7-s + (−0.790 + 0.611i)8-s + (−0.114 + 0.199i)9-s + (−0.461 − 0.501i)10-s + 0.0634i·11-s + (−0.501 + 0.720i)12-s + (−0.126 − 0.219i)13-s + (1.20 + 1.31i)14-s + (−0.518 − 0.299i)15-s + (0.638 − 0.769i)16-s + (−0.689 − 0.397i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0965 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0965 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(296\)    =    \(2^{3} \cdot 37\)
Sign: $-0.0965 + 0.995i$
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: \((2,\ 296,\ (\ :1/2),\ -0.0965 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.202987 - 0.223623i\)
\(L(\frac12)\) \(\approx\) \(0.202987 - 0.223623i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.38 - 0.307i)T \)
37 \( 1 + (3.36 + 5.06i)T \)
good3 \( 1 + (1.31 - 0.759i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.762 - 1.31i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (2.35 + 4.08i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 0.210iT - 11T^{2} \)
13 \( 1 + (0.457 + 0.792i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.84 + 1.64i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.642 + 1.11i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 6.24iT - 23T^{2} \)
29 \( 1 + 1.76T + 29T^{2} \)
31 \( 1 + 2.65iT - 31T^{2} \)
41 \( 1 + (1.18 + 2.04i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 6.62T + 43T^{2} \)
47 \( 1 - 11.5T + 47T^{2} \)
53 \( 1 + (-8.29 - 4.78i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.93 - 10.2i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.99 + 5.18i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (11.0 - 6.36i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.90 + 5.02i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 4.74T + 73T^{2} \)
79 \( 1 + (8.93 - 5.15i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.68 + 5.01i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.86 + 2.23i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 3.33iT - 97T^{2} \)
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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

−10.89815906828671882439799230167, −10.55608090009847743166746090803, −9.976625203970552677439421758803, −8.814702938158602474575802752452, −7.41613447441484433394381061573, −6.75319838454432624106196443066, −5.86193783317377054395023182623, −4.35703957228092775788415901485, −2.65953890440835640612359295888, −0.31859396652935341464563933252, 1.71572122181301092553162955946, 3.20661642560262068914979549464, 5.45663747257275467563700538631, 6.15684389046277755297035654069, 7.06148121546636202382479626892, 8.610483298743206887832054591780, 9.092552128876373175242845437415, 9.946511250970417384730893062010, 11.24072452399823836544883250704, 11.97221069363985889827425987985

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