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Properties

Label 2-237-237.236-c1-0-20
Degree $2$
Conductor $237$
Sign $0.733 + 0.679i$
Analytic cond. $1.89245$
Root an. cond. $1.37566$
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  + 0.221i·2-s + (1.11 − 1.32i)3-s + 1.95·4-s − 3.46i·5-s + (0.294 + 0.245i)6-s + 4.18i·7-s + 0.874i·8-s + (−0.531 − 2.95i)9-s + 0.765·10-s + 0.891i·11-s + (2.16 − 2.59i)12-s − 4.83·13-s − 0.925·14-s + (−4.59 − 3.84i)15-s + 3.70·16-s − 4.42·17-s + ⋯
L(s)  = 1  + 0.156i·2-s + (0.641 − 0.767i)3-s + 0.975·4-s − 1.54i·5-s + (0.120 + 0.100i)6-s + 1.58i·7-s + 0.309i·8-s + (−0.177 − 0.984i)9-s + 0.242·10-s + 0.268i·11-s + (0.625 − 0.748i)12-s − 1.33·13-s − 0.247·14-s + (−1.18 − 0.992i)15-s + 0.927·16-s − 1.07·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 237 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.733 + 0.679i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 237 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.733 + 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(237\)    =    \(3 \cdot 79\)
Sign: $0.733 + 0.679i$
Analytic conductor: \(1.89245\)
Root analytic conductor: \(1.37566\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{237} (236, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 237,\ (\ :1/2),\ 0.733 + 0.679i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.60619 - 0.629112i\)
\(L(\frac12)\) \(\approx\) \(1.60619 - 0.629112i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.11 + 1.32i)T \)
79 \( 1 + (-0.446 + 8.87i)T \)
good2 \( 1 - 0.221iT - 2T^{2} \)
5 \( 1 + 3.46iT - 5T^{2} \)
7 \( 1 - 4.18iT - 7T^{2} \)
11 \( 1 - 0.891iT - 11T^{2} \)
13 \( 1 + 4.83T + 13T^{2} \)
17 \( 1 + 4.42T + 17T^{2} \)
19 \( 1 - 3.83T + 19T^{2} \)
23 \( 1 - 6.53iT - 23T^{2} \)
29 \( 1 - 7.65T + 29T^{2} \)
31 \( 1 - 3.13T + 31T^{2} \)
37 \( 1 + 7.65iT - 37T^{2} \)
41 \( 1 + 9.46T + 41T^{2} \)
43 \( 1 - 5.67iT - 43T^{2} \)
47 \( 1 + 1.80T + 47T^{2} \)
53 \( 1 - 8.50T + 53T^{2} \)
59 \( 1 + 9.93T + 59T^{2} \)
61 \( 1 - 4.69iT - 61T^{2} \)
67 \( 1 + 3.58T + 67T^{2} \)
71 \( 1 + 2.11T + 71T^{2} \)
73 \( 1 - 3.44T + 73T^{2} \)
83 \( 1 + 5.06iT - 83T^{2} \)
89 \( 1 - 7.84iT - 89T^{2} \)
97 \( 1 + 7.89T + 97T^{2} \)
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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

−12.05455606420269369274481775223, −11.74077705152077098125857135947, −9.710444907131196377762556249776, −8.950561596463723360945776547052, −8.143979362575917425206585636718, −7.16042809630761922803632427985, −5.90823067867876959506169272550, −4.94739248397548356482573190369, −2.78199389253956246683728751616, −1.74033064099237928273262749499, 2.50870401536074509564518608104, 3.32519331799289595911085718340, 4.62184312722732578725602480181, 6.65628107394341635281304399303, 7.10835914922670913190742528736, 8.131899444611225691112555284212, 9.960946034519642216738488257111, 10.34793365691066612007597405738, 10.94740930750243884313146274020, 11.93384985568938203508327103842

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