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Properties

Label 2-296-296.101-c1-0-17
Degree $2$
Conductor $296$
Sign $0.985 - 0.167i$
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  + (−1.33 + 0.467i)2-s + (2.44 − 1.41i)3-s + (1.56 − 1.24i)4-s + (0.380 + 0.658i)5-s + (−2.60 + 3.02i)6-s + (1.51 + 2.62i)7-s + (−1.50 + 2.39i)8-s + (2.48 − 4.29i)9-s + (−0.815 − 0.701i)10-s + 4.75i·11-s + (2.05 − 5.25i)12-s + (2.66 + 4.61i)13-s + (−3.25 − 2.79i)14-s + (1.85 + 1.07i)15-s + (0.884 − 3.90i)16-s + (−3.24 − 1.87i)17-s + ⋯
L(s)  = 1  + (−0.943 + 0.330i)2-s + (1.41 − 0.814i)3-s + (0.781 − 0.624i)4-s + (0.170 + 0.294i)5-s + (−1.06 + 1.23i)6-s + (0.573 + 0.992i)7-s + (−0.531 + 0.847i)8-s + (0.826 − 1.43i)9-s + (−0.257 − 0.221i)10-s + 1.43i·11-s + (0.594 − 1.51i)12-s + (0.739 + 1.28i)13-s + (−0.869 − 0.747i)14-s + (0.479 + 0.276i)15-s + (0.221 − 0.975i)16-s + (−0.787 − 0.454i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.167i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 - 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(296\)    =    \(2^{3} \cdot 37\)
Sign: $0.985 - 0.167i$
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: \((2,\ 296,\ (\ :1/2),\ 0.985 - 0.167i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42187 + 0.120264i\)
\(L(\frac12)\) \(\approx\) \(1.42187 + 0.120264i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.33 - 0.467i)T \)
37 \( 1 + (5.34 - 2.91i)T \)
good3 \( 1 + (-2.44 + 1.41i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.380 - 0.658i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.51 - 2.62i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 4.75iT - 11T^{2} \)
13 \( 1 + (-2.66 - 4.61i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.24 + 1.87i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.18 + 7.24i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 6.95iT - 23T^{2} \)
29 \( 1 + 5.45T + 29T^{2} \)
31 \( 1 + 2.47iT - 31T^{2} \)
41 \( 1 + (1.20 + 2.08i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 3.43T + 43T^{2} \)
47 \( 1 + 0.977T + 47T^{2} \)
53 \( 1 + (3.03 + 1.75i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.51 - 2.62i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.770 - 1.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-13.2 + 7.62i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.41 + 2.44i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 7.88T + 73T^{2} \)
79 \( 1 + (-0.687 + 0.396i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.50 - 1.44i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-3.62 - 2.09i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 9.59iT - 97T^{2} \)
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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

−11.74705629926185351894350013456, −10.72606175026564030295489661501, −9.275662737197742780079834856790, −8.949679090659188904688751489520, −8.179210857507413860141083491510, −6.94105375572816574274531503961, −6.60675628119988365644410105447, −4.63807586009756837259081775497, −2.36866845725502318001338429625, −2.06662565625374054339054013223, 1.56963111349864104706122462596, 3.34258880738522461046305335575, 3.86138648548231792000478497026, 5.79009825529545570039959195160, 7.52278398685845906760401078811, 8.316243296284632319462395503613, 8.715011334866553921353282018337, 9.814163405886752161302856089416, 10.70277972363419773350190554553, 11.08609304305303520339163532027

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