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Properties

Label 2-37-37.11-c5-0-9
Degree $2$
Conductor $37$
Sign $0.688 + 0.725i$
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  + (−2.20 + 1.27i)2-s + (14.9 − 25.8i)3-s + (−12.7 + 22.0i)4-s + (67.9 + 39.2i)5-s + 76.0i·6-s + (34.5 − 59.7i)7-s − 146. i·8-s + (−322. − 559. i)9-s − 200.·10-s + 667.·11-s + (379. + 658. i)12-s + (−513. − 296. i)13-s + 175. i·14-s + (2.02e3 − 1.16e3i)15-s + (−220. − 382. i)16-s + (−354. + 204. i)17-s + ⋯
L(s)  = 1  + (−0.390 + 0.225i)2-s + (0.956 − 1.65i)3-s + (−0.398 + 0.689i)4-s + (1.21 + 0.701i)5-s + 0.862i·6-s + (0.266 − 0.460i)7-s − 0.810i·8-s + (−1.32 − 2.30i)9-s − 0.632·10-s + 1.66·11-s + (0.761 + 1.31i)12-s + (−0.843 − 0.486i)13-s + 0.239i·14-s + (2.32 − 1.34i)15-s + (−0.215 − 0.373i)16-s + (−0.297 + 0.171i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.688 + 0.725i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.688 + 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.688 + 0.725i$
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: \((2,\ 37,\ (\ :5/2),\ 0.688 + 0.725i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.75481 - 0.754126i\)
\(L(\frac12)\) \(\approx\) \(1.75481 - 0.754126i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (4.58e3 - 6.95e3i)T \)
good2 \( 1 + (2.20 - 1.27i)T + (16 - 27.7i)T^{2} \)
3 \( 1 + (-14.9 + 25.8i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (-67.9 - 39.2i)T + (1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + (-34.5 + 59.7i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 - 667.T + 1.61e5T^{2} \)
13 \( 1 + (513. + 296. i)T + (1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 + (354. - 204. i)T + (7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-1.10e3 - 637. i)T + (1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + 463. iT - 6.43e6T^{2} \)
29 \( 1 - 4.94e3iT - 2.05e7T^{2} \)
31 \( 1 - 3.67e3iT - 2.86e7T^{2} \)
41 \( 1 + (1.94e3 - 3.37e3i)T + (-5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 - 1.64e3iT - 1.47e8T^{2} \)
47 \( 1 + 9.50e3T + 2.29e8T^{2} \)
53 \( 1 + (1.03e4 + 1.79e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (1.05e4 - 6.08e3i)T + (3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-1.78e4 - 1.03e4i)T + (4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-1.73e4 + 3.01e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + (1.85e4 - 3.20e4i)T + (-9.02e8 - 1.56e9i)T^{2} \)
73 \( 1 + 3.22e4T + 2.07e9T^{2} \)
79 \( 1 + (-1.34e4 - 7.78e3i)T + (1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (6.03e4 + 1.04e5i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 + (6.71e4 - 3.87e4i)T + (2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + 6.23e4iT - 8.58e9T^{2} \)
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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

−14.50086824436077126091834353172, −14.07268862022782529905961061327, −13.03382681127119559169846705014, −11.98970835104735369472207770199, −9.706847959111316726567698288012, −8.629352065898757674226007947490, −7.29431835600490070699615426285, −6.53426471809141556500412559545, −3.16512106426709090666825927169, −1.43865451368995882955134594406, 2.04988412751914286744004871818, 4.43099835409409561656063781285, 5.54105579703798004152593042026, 8.752555247762282374421840037025, 9.398068833815354453436868691688, 9.862900065944842001007395290645, 11.43603729678546198998304364879, 13.76740453652096300760522267593, 14.32824783613460118666427528224, 15.30479886163276747153439382706

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