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Properties

Label 2-37-37.10-c7-0-4
Degree $2$
Conductor $37$
Sign $0.992 + 0.118i$
Analytic cond. $11.5582$
Root an. cond. $3.39974$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.32 − 4.01i)2-s + (−46.0 − 79.8i)3-s + (53.2 + 92.1i)4-s + (217. + 376. i)5-s − 427.·6-s + (19.1 + 33.0i)7-s + 1.08e3·8-s + (−3.15e3 + 5.46e3i)9-s + 2.01e3·10-s + 3.75e3·11-s + (4.90e3 − 8.50e3i)12-s + (−378. − 655. i)13-s + 177.·14-s + (2.00e4 − 3.47e4i)15-s + (−4.28e3 + 7.42e3i)16-s + (1.29e4 − 2.24e4i)17-s + ⋯
L(s)  = 1  + (0.205 − 0.355i)2-s + (−0.985 − 1.70i)3-s + (0.415 + 0.720i)4-s + (0.777 + 1.34i)5-s − 0.808·6-s + (0.0210 + 0.0364i)7-s + 0.751·8-s + (−1.44 + 2.49i)9-s + 0.637·10-s + 0.850·11-s + (0.819 − 1.41i)12-s + (−0.0477 − 0.0827i)13-s + 0.0172·14-s + (1.53 − 2.65i)15-s + (−0.261 + 0.453i)16-s + (0.640 − 1.10i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.118i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.992 + 0.118i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.992 + 0.118i$
Analytic conductor: \(11.5582\)
Root analytic conductor: \(3.39974\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :7/2),\ 0.992 + 0.118i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.80618 - 0.107295i\)
\(L(\frac12)\) \(\approx\) \(1.80618 - 0.107295i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (-7.85e4 - 2.97e5i)T \)
good2 \( 1 + (-2.32 + 4.01i)T + (-64 - 110. i)T^{2} \)
3 \( 1 + (46.0 + 79.8i)T + (-1.09e3 + 1.89e3i)T^{2} \)
5 \( 1 + (-217. - 376. i)T + (-3.90e4 + 6.76e4i)T^{2} \)
7 \( 1 + (-19.1 - 33.0i)T + (-4.11e5 + 7.13e5i)T^{2} \)
11 \( 1 - 3.75e3T + 1.94e7T^{2} \)
13 \( 1 + (378. + 655. i)T + (-3.13e7 + 5.43e7i)T^{2} \)
17 \( 1 + (-1.29e4 + 2.24e4i)T + (-2.05e8 - 3.55e8i)T^{2} \)
19 \( 1 + (-2.28e4 - 3.95e4i)T + (-4.46e8 + 7.74e8i)T^{2} \)
23 \( 1 + 3.28e4T + 3.40e9T^{2} \)
29 \( 1 + 3.60e4T + 1.72e10T^{2} \)
31 \( 1 - 1.05e5T + 2.75e10T^{2} \)
41 \( 1 + (4.03e4 + 6.98e4i)T + (-9.73e10 + 1.68e11i)T^{2} \)
43 \( 1 - 7.31e4T + 2.71e11T^{2} \)
47 \( 1 - 5.97e5T + 5.06e11T^{2} \)
53 \( 1 + (4.54e4 - 7.87e4i)T + (-5.87e11 - 1.01e12i)T^{2} \)
59 \( 1 + (1.34e6 - 2.33e6i)T + (-1.24e12 - 2.15e12i)T^{2} \)
61 \( 1 + (6.88e5 + 1.19e6i)T + (-1.57e12 + 2.72e12i)T^{2} \)
67 \( 1 + (7.53e5 + 1.30e6i)T + (-3.03e12 + 5.24e12i)T^{2} \)
71 \( 1 + (-2.30e6 - 3.99e6i)T + (-4.54e12 + 7.87e12i)T^{2} \)
73 \( 1 + 3.68e6T + 1.10e13T^{2} \)
79 \( 1 + (2.38e6 + 4.12e6i)T + (-9.60e12 + 1.66e13i)T^{2} \)
83 \( 1 + (-3.58e6 + 6.20e6i)T + (-1.35e13 - 2.35e13i)T^{2} \)
89 \( 1 + (2.55e5 - 4.42e5i)T + (-2.21e13 - 3.83e13i)T^{2} \)
97 \( 1 + 8.13e6T + 8.07e13T^{2} \)
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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

−14.19088321169256072332068334356, −13.54994122620247694662870520568, −12.11360234846754738674423197193, −11.63373693463521635562827594927, −10.37729222932503743773300590646, −7.72927940052676681792878603982, −6.88484992769217331718634073250, −5.87387918529798938186677341235, −2.82676636745706362773448561416, −1.54391137394389228192634728050, 0.954379756841419345975421206755, 4.31848086107822193408893073955, 5.34321283030670572949835277343, 6.16693918458789849039186212247, 9.104085351288348498040017235211, 9.837010836113021528703308402614, 10.99482393118530213803677156047, 12.20650567416436741339416287721, 14.04070641634979357457778263056, 15.20655438213525552366032602472

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