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Properties

Label 2-37-37.11-c7-0-14
Degree $2$
Conductor $37$
Sign $0.776 + 0.629i$
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  + (−14.9 + 8.62i)2-s + (30.8 − 53.5i)3-s + (84.6 − 146. i)4-s + (237. + 136. i)5-s + 1.06e3i·6-s + (106. − 184. i)7-s + 712. i·8-s + (−815. − 1.41e3i)9-s − 4.72e3·10-s − 1.46e3·11-s + (−5.23e3 − 9.05e3i)12-s + (−2.64e3 − 1.52e3i)13-s + 3.67e3i·14-s + (1.46e4 − 8.46e3i)15-s + (4.69e3 + 8.12e3i)16-s + (2.55e4 − 1.47e4i)17-s + ⋯
L(s)  = 1  + (−1.31 + 0.762i)2-s + (0.660 − 1.14i)3-s + (0.661 − 1.14i)4-s + (0.848 + 0.489i)5-s + 2.01i·6-s + (0.117 − 0.203i)7-s + 0.492i·8-s + (−0.372 − 0.645i)9-s − 1.49·10-s − 0.332·11-s + (−0.873 − 1.51i)12-s + (−0.334 − 0.193i)13-s + 0.358i·14-s + (1.12 − 0.647i)15-s + (0.286 + 0.496i)16-s + (1.26 − 0.727i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.776 + 0.629i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.776 + 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.776 + 0.629i$
Analytic conductor: \(11.5582\)
Root analytic conductor: \(3.39974\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :7/2),\ 0.776 + 0.629i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.17840 - 0.417623i\)
\(L(\frac12)\) \(\approx\) \(1.17840 - 0.417623i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (-1.34e5 + 2.77e5i)T \)
good2 \( 1 + (14.9 - 8.62i)T + (64 - 110. i)T^{2} \)
3 \( 1 + (-30.8 + 53.5i)T + (-1.09e3 - 1.89e3i)T^{2} \)
5 \( 1 + (-237. - 136. i)T + (3.90e4 + 6.76e4i)T^{2} \)
7 \( 1 + (-106. + 184. i)T + (-4.11e5 - 7.13e5i)T^{2} \)
11 \( 1 + 1.46e3T + 1.94e7T^{2} \)
13 \( 1 + (2.64e3 + 1.52e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
17 \( 1 + (-2.55e4 + 1.47e4i)T + (2.05e8 - 3.55e8i)T^{2} \)
19 \( 1 + (-1.37e4 - 7.91e3i)T + (4.46e8 + 7.74e8i)T^{2} \)
23 \( 1 + 7.34e4iT - 3.40e9T^{2} \)
29 \( 1 + 6.07e4iT - 1.72e10T^{2} \)
31 \( 1 + 1.21e5iT - 2.75e10T^{2} \)
41 \( 1 + (-1.96e5 + 3.40e5i)T + (-9.73e10 - 1.68e11i)T^{2} \)
43 \( 1 - 3.39e5iT - 2.71e11T^{2} \)
47 \( 1 - 3.24e5T + 5.06e11T^{2} \)
53 \( 1 + (4.25e5 + 7.36e5i)T + (-5.87e11 + 1.01e12i)T^{2} \)
59 \( 1 + (2.32e6 - 1.34e6i)T + (1.24e12 - 2.15e12i)T^{2} \)
61 \( 1 + (1.71e5 + 9.91e4i)T + (1.57e12 + 2.72e12i)T^{2} \)
67 \( 1 + (1.80e6 - 3.12e6i)T + (-3.03e12 - 5.24e12i)T^{2} \)
71 \( 1 + (-1.04e5 + 1.80e5i)T + (-4.54e12 - 7.87e12i)T^{2} \)
73 \( 1 - 3.43e6T + 1.10e13T^{2} \)
79 \( 1 + (-6.56e6 - 3.79e6i)T + (9.60e12 + 1.66e13i)T^{2} \)
83 \( 1 + (1.55e6 + 2.69e6i)T + (-1.35e13 + 2.35e13i)T^{2} \)
89 \( 1 + (3.11e6 - 1.80e6i)T + (2.21e13 - 3.83e13i)T^{2} \)
97 \( 1 + 1.25e7iT - 8.07e13T^{2} \)
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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

−14.64751503331487050074280386125, −13.81599951682735328781674098094, −12.49736270044189328239689213002, −10.45316445174570659775689154340, −9.422019323667092402706461225565, −7.990141347407802673207117842340, −7.29438077201848865165991630925, −6.02744387353920079759544583814, −2.44122643673523290898555334475, −0.876940316216398100423747718364, 1.50046102050850185463181902925, 3.15954892139612675334964780557, 5.25380354302134533276067034727, 7.956870074069771137712293208155, 9.202685824440501643023863952135, 9.717073262205862992998150944741, 10.69069272141862788038265819033, 12.21317764917723407200800058658, 13.88531021448136050117840771461, 15.15482758849075716022429613870

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