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Properties

Label 2-37-37.27-c5-0-1
Degree $2$
Conductor $37$
Sign $-0.999 + 0.00955i$
Analytic cond. $5.93420$
Root an. cond. $2.43602$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.312 + 0.180i)2-s + (9.26 + 16.0i)3-s + (−15.9 − 27.6i)4-s + (−90.8 + 52.4i)5-s + 6.68i·6-s + (−48.1 − 83.3i)7-s − 23.0i·8-s + (−50.2 + 86.9i)9-s − 37.8·10-s − 46.1·11-s + (295. − 511. i)12-s + (−659. + 380. i)13-s − 34.7i·14-s + (−1.68e3 − 971. i)15-s + (−505. + 876. i)16-s + (1.26e3 + 729. i)17-s + ⋯
L(s)  = 1  + (0.0552 + 0.0318i)2-s + (0.594 + 1.02i)3-s + (−0.497 − 0.862i)4-s + (−1.62 + 0.937i)5-s + 0.0757i·6-s + (−0.371 − 0.643i)7-s − 0.127i·8-s + (−0.206 + 0.357i)9-s − 0.119·10-s − 0.114·11-s + (0.591 − 1.02i)12-s + (−1.08 + 0.624i)13-s − 0.0473i·14-s + (−1.93 − 1.11i)15-s + (−0.493 + 0.855i)16-s + (1.06 + 0.612i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00955i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.00955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $-0.999 + 0.00955i$
Analytic conductor: \(5.93420\)
Root analytic conductor: \(2.43602\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :5/2),\ -0.999 + 0.00955i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.00188029 - 0.393389i\)
\(L(\frac12)\) \(\approx\) \(0.00188029 - 0.393389i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (1.81e3 - 8.12e3i)T \)
good2 \( 1 + (-0.312 - 0.180i)T + (16 + 27.7i)T^{2} \)
3 \( 1 + (-9.26 - 16.0i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (90.8 - 52.4i)T + (1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (48.1 + 83.3i)T + (-8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + 46.1T + 1.61e5T^{2} \)
13 \( 1 + (659. - 380. i)T + (1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 + (-1.26e3 - 729. i)T + (7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (1.82e3 - 1.05e3i)T + (1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + 393. iT - 6.43e6T^{2} \)
29 \( 1 + 1.47e3iT - 2.05e7T^{2} \)
31 \( 1 - 6.78e3iT - 2.86e7T^{2} \)
41 \( 1 + (2.52e3 + 4.37e3i)T + (-5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 + 1.60e4iT - 1.47e8T^{2} \)
47 \( 1 + 1.93e4T + 2.29e8T^{2} \)
53 \( 1 + (-305. + 528. i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (5.85e3 + 3.37e3i)T + (3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (3.35e4 - 1.93e4i)T + (4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (1.61e4 + 2.78e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + (-2.42e4 - 4.20e4i)T + (-9.02e8 + 1.56e9i)T^{2} \)
73 \( 1 + 1.34e4T + 2.07e9T^{2} \)
79 \( 1 + (5.24e4 - 3.03e4i)T + (1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (2.60e4 - 4.50e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 + (-4.41e3 - 2.54e3i)T + (2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 7.15e4iT - 8.58e9T^{2} \)
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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

−15.53893165925899387166712132969, −14.81392681916419675024560516021, −14.23336715701105368748280593361, −12.22192258572887031171378128571, −10.58582364939769346537844106782, −10.03628557194303014157958147442, −8.411256362831286878639002037032, −6.86549174953038786190452438825, −4.47449778140903098203944839667, −3.52764027235220321371672701184, 0.20691829957368681023509797199, 2.98582154202415582193324403624, 4.71012319958066277543365838134, 7.47373456463356418996232810734, 8.027846456810824219041446599532, 9.119101683054970002207954093650, 11.74188230404862878399033612304, 12.59597065737066853634659448804, 13.01643486427185055173762842807, 14.74497271155099945094558524549

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