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Properties

Label 2-237-237.173-c3-0-20
Degree $2$
Conductor $237$
Sign $-0.215 - 0.976i$
Analytic cond. $13.9834$
Root an. cond. $3.73944$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.27 − 2.95i)3-s + (0.964 + 7.94i)4-s + (−20.9 + 14.4i)7-s + (9.57 − 25.2i)9-s + (27.5 + 31.1i)12-s + (10.0 + 82.5i)13-s + (−62.1 + 15.3i)16-s + (−85.4 + 44.8i)19-s + (−46.9 + 123. i)21-s + (110. + 58.0i)25-s + (−33.5 − 136. i)27-s + (−135. − 152. i)28-s + (−88.0 − 77.9i)31-s + (209. + 51.6i)36-s + (185. + 353. i)37-s + ⋯
L(s)  = 1  + (0.822 − 0.568i)3-s + (0.120 + 0.992i)4-s + (−1.13 + 0.781i)7-s + (0.354 − 0.935i)9-s + (0.663 + 0.748i)12-s + (0.213 + 1.76i)13-s + (−0.970 + 0.239i)16-s + (−1.03 + 0.541i)19-s + (−0.487 + 1.28i)21-s + (0.885 + 0.464i)25-s + (−0.239 − 0.970i)27-s + (−0.912 − 1.02i)28-s + (−0.510 − 0.451i)31-s + (0.970 + 0.239i)36-s + (0.825 + 1.57i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 237 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.215 - 0.976i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 237 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.215 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(237\)    =    \(3 \cdot 79\)
Sign: $-0.215 - 0.976i$
Analytic conductor: \(13.9834\)
Root analytic conductor: \(3.73944\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{237} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 237,\ (\ :3/2),\ -0.215 - 0.976i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.03843 + 1.29193i\)
\(L(\frac12)\) \(\approx\) \(1.03843 + 1.29193i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.27 + 2.95i)T \)
79 \( 1 + (-559. - 423. i)T \)
good2 \( 1 + (-0.964 - 7.94i)T^{2} \)
5 \( 1 + (-110. - 58.0i)T^{2} \)
7 \( 1 + (20.9 - 14.4i)T + (121. - 320. i)T^{2} \)
11 \( 1 + (-1.17e3 + 618. i)T^{2} \)
13 \( 1 + (-10.0 - 82.5i)T + (-2.13e3 + 525. i)T^{2} \)
17 \( 1 + (-4.77e3 + 1.17e3i)T^{2} \)
19 \( 1 + (85.4 - 44.8i)T + (3.89e3 - 5.64e3i)T^{2} \)
23 \( 1 - 1.21e4T^{2} \)
29 \( 1 + (-1.82e4 + 1.61e4i)T^{2} \)
31 \( 1 + (88.0 + 77.9i)T + (3.59e3 + 2.95e4i)T^{2} \)
37 \( 1 + (-185. - 353. i)T + (-2.87e4 + 4.16e4i)T^{2} \)
41 \( 1 + (6.10e4 + 3.20e4i)T^{2} \)
43 \( 1 + (11.5 - 47.0i)T + (-7.03e4 - 3.69e4i)T^{2} \)
47 \( 1 + (5.89e4 + 8.54e4i)T^{2} \)
53 \( 1 + (-5.27e4 - 1.39e5i)T^{2} \)
59 \( 1 + (-1.99e5 - 4.91e4i)T^{2} \)
61 \( 1 + (-31.5 - 60.1i)T + (-1.28e5 + 1.86e5i)T^{2} \)
67 \( 1 + (-818. + 724. i)T + (3.62e4 - 2.98e5i)T^{2} \)
71 \( 1 + (-1.26e5 - 3.34e5i)T^{2} \)
73 \( 1 + (77.4 - 637. i)T + (-3.77e5 - 9.30e4i)T^{2} \)
83 \( 1 + (5.55e5 - 1.36e5i)T^{2} \)
89 \( 1 + (2.49e5 - 6.59e5i)T^{2} \)
97 \( 1 + (-1.60e3 + 843. i)T + (5.18e5 - 7.51e5i)T^{2} \)
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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

−12.25008723740798998738934654772, −11.33962900906625989813478326103, −9.616198374767902525845794699656, −8.961829566485660723545720166084, −8.172012962579502412392664745268, −6.89043929521817099758082803603, −6.35929412996753486515001978432, −4.19303773258311620368252341171, −3.13714957995818352112565473481, −2.05297700589423408146002986523, 0.58098417744964130779991664520, 2.57750831966636593541570548708, 3.77920292135288967505514249992, 5.09728353543120641021565659900, 6.31896090160547035878354846352, 7.43294655811460266091196493182, 8.702938401049476347713026252751, 9.638240120419526347908282151514, 10.50532918149931206933161519866, 10.78998469112396485381233859686

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