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Properties

Label 2-37-37.11-c5-0-7
Degree $2$
Conductor $37$
Sign $0.832 - 0.553i$
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  + (4.33 − 2.50i)2-s + (−3.26 + 5.66i)3-s + (−3.48 + 6.03i)4-s + (54.4 + 31.4i)5-s + 32.7i·6-s + (22.9 − 39.6i)7-s + 194. i·8-s + (100. + 173. i)9-s + 314.·10-s + 436.·11-s + (−22.7 − 39.4i)12-s + (−639. − 369. i)13-s − 229. i·14-s + (−355. + 205. i)15-s + (376. + 651. i)16-s + (979. − 565. i)17-s + ⋯
L(s)  = 1  + (0.765 − 0.442i)2-s + (−0.209 + 0.363i)3-s + (−0.108 + 0.188i)4-s + (0.973 + 0.561i)5-s + 0.371i·6-s + (0.176 − 0.306i)7-s + 1.07i·8-s + (0.412 + 0.713i)9-s + 0.993·10-s + 1.08·11-s + (−0.0456 − 0.0790i)12-s + (−1.04 − 0.605i)13-s − 0.312i·14-s + (−0.408 + 0.235i)15-s + (0.367 + 0.636i)16-s + (0.822 − 0.474i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.832 - 0.553i$
Analytic conductor: \(5.93420\)
Root analytic conductor: \(2.43602\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :5/2),\ 0.832 - 0.553i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.27753 + 0.688086i\)
\(L(\frac12)\) \(\approx\) \(2.27753 + 0.688086i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (-6.06e3 - 5.70e3i)T \)
good2 \( 1 + (-4.33 + 2.50i)T + (16 - 27.7i)T^{2} \)
3 \( 1 + (3.26 - 5.66i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (-54.4 - 31.4i)T + (1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + (-22.9 + 39.6i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 - 436.T + 1.61e5T^{2} \)
13 \( 1 + (639. + 369. i)T + (1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 + (-979. + 565. i)T + (7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (1.73e3 + 9.99e2i)T + (1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 - 2.73e3iT - 6.43e6T^{2} \)
29 \( 1 + 564. iT - 2.05e7T^{2} \)
31 \( 1 + 1.02e4iT - 2.86e7T^{2} \)
41 \( 1 + (-3.85e3 + 6.67e3i)T + (-5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + 1.67e4iT - 1.47e8T^{2} \)
47 \( 1 - 1.94e4T + 2.29e8T^{2} \)
53 \( 1 + (-1.32e3 - 2.29e3i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (6.87e3 - 3.96e3i)T + (3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (1.69e3 + 976. i)T + (4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-8.23e3 + 1.42e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + (-1.62e3 + 2.80e3i)T + (-9.02e8 - 1.56e9i)T^{2} \)
73 \( 1 + 4.95e4T + 2.07e9T^{2} \)
79 \( 1 + (3.88e4 + 2.24e4i)T + (1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (-4.92e3 - 8.52e3i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 + (4.03e4 - 2.33e4i)T + (2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 1.15e5iT - 8.58e9T^{2} \)
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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

−15.08654267142918738206465102759, −14.05160786721738749064493044196, −13.24736908152578873621440700690, −11.89604398643450987729899640042, −10.66610822295652224462864373147, −9.515852745228516565841411592909, −7.53267812861197723859613301503, −5.63480502874963407679566298743, −4.23485778380024168879538425897, −2.36789764632263059888040423324, 1.38709044426868978066750072202, 4.34409783082056496569318672838, 5.81785589564648242635494097818, 6.76685021958613034088015199032, 9.043458910785964115325550316718, 10.04388472920279748588388206667, 12.23687522376553416137202554611, 12.80375090162915153256320346525, 14.32628228487524673583666090231, 14.74196887311460684346901413997

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