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Properties

Label 2-237-237.41-c3-0-69
Degree $2$
Conductor $237$
Sign $0.101 + 0.994i$
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  + (5.15 − 0.626i)3-s + (4.54 − 6.58i)4-s + (−29.5 + 3.58i)7-s + (26.2 − 6.46i)9-s + (19.3 − 36.8i)12-s + (47.0 − 68.2i)13-s + (−22.6 − 59.8i)16-s + (110. − 98.1i)19-s + (−150. + 37.0i)21-s + (−93.5 − 82.8i)25-s + (131. − 49.7i)27-s + (−110. + 210. i)28-s + (−268. + 140. i)31-s + (76.5 − 201. i)36-s + (222. + 251. i)37-s + ⋯
L(s)  = 1  + (0.992 − 0.120i)3-s + (0.568 − 0.822i)4-s + (−1.59 + 0.193i)7-s + (0.970 − 0.239i)9-s + (0.464 − 0.885i)12-s + (1.00 − 1.45i)13-s + (−0.354 − 0.935i)16-s + (1.33 − 1.18i)19-s + (−1.55 + 0.384i)21-s + (−0.748 − 0.663i)25-s + (0.935 − 0.354i)27-s + (−0.746 + 1.42i)28-s + (−1.55 + 0.816i)31-s + (0.354 − 0.935i)36-s + (0.988 + 1.11i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 237 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.101 + 0.994i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 237 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.101 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(237\)    =    \(3 \cdot 79\)
Sign: $0.101 + 0.994i$
Analytic conductor: \(13.9834\)
Root analytic conductor: \(3.73944\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{237} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 237,\ (\ :3/2),\ 0.101 + 0.994i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.79255 - 1.61845i\)
\(L(\frac12)\) \(\approx\) \(1.79255 - 1.61845i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-5.15 + 0.626i)T \)
79 \( 1 + (353. - 606. i)T \)
good2 \( 1 + (-4.54 + 6.58i)T^{2} \)
5 \( 1 + (93.5 + 82.8i)T^{2} \)
7 \( 1 + (29.5 - 3.58i)T + (333. - 82.0i)T^{2} \)
11 \( 1 + (996. - 882. i)T^{2} \)
13 \( 1 + (-47.0 + 68.2i)T + (-779. - 2.05e3i)T^{2} \)
17 \( 1 + (-1.74e3 - 4.59e3i)T^{2} \)
19 \( 1 + (-110. + 98.1i)T + (826. - 6.80e3i)T^{2} \)
23 \( 1 - 1.21e4T^{2} \)
29 \( 1 + (2.15e4 + 1.13e4i)T^{2} \)
31 \( 1 + (268. - 140. i)T + (1.69e4 - 2.45e4i)T^{2} \)
37 \( 1 + (-222. - 251. i)T + (-6.10e3 + 5.02e4i)T^{2} \)
41 \( 1 + (-5.15e4 - 4.57e4i)T^{2} \)
43 \( 1 + (169. + 64.3i)T + (5.95e4 + 5.27e4i)T^{2} \)
47 \( 1 + (1.25e4 + 1.03e5i)T^{2} \)
53 \( 1 + (-1.44e5 - 3.56e4i)T^{2} \)
59 \( 1 + (-7.28e4 + 1.92e5i)T^{2} \)
61 \( 1 + (-451. - 509. i)T + (-2.73e4 + 2.25e5i)T^{2} \)
67 \( 1 + (-420. - 220. i)T + (1.70e5 + 2.47e5i)T^{2} \)
71 \( 1 + (-3.47e5 - 8.56e4i)T^{2} \)
73 \( 1 + (-697. - 1.01e3i)T + (-1.37e5 + 3.63e5i)T^{2} \)
83 \( 1 + (2.02e5 + 5.34e5i)T^{2} \)
89 \( 1 + (6.84e5 - 1.68e5i)T^{2} \)
97 \( 1 + (-1.42e3 + 1.26e3i)T + (1.10e5 - 9.06e5i)T^{2} \)
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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

−11.40221787132456544725172951321, −10.18645706190145128656682396981, −9.691642081548910722725010841567, −8.676027197851856896887712853413, −7.37055585326941968202519866756, −6.48368308785553226411833735249, −5.44984941381241513550963394879, −3.48229715704420736663423549989, −2.67396302491468354222439159924, −0.880898730821816239012728779246, 1.94802881858564279571639666081, 3.47617774370563333735753648968, 3.80042859166343207258647310784, 6.11113841912756897964379792565, 7.08269638732934997523022763020, 7.892342383385639773420389953990, 9.189068673965156965938891136382, 9.637204851336089534862452252663, 11.00538965644459869236607601367, 12.06128134457152692536243150375

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