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Properties

Label 2-37-37.11-c5-0-1
Degree $2$
Conductor $37$
Sign $-0.497 - 0.867i$
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  + (5.44 − 3.14i)2-s + (−10.4 + 18.1i)3-s + (3.74 − 6.49i)4-s + (−46.0 − 26.6i)5-s + 131. i·6-s + (−56.9 + 98.6i)7-s + 154. i·8-s + (−98.1 − 170. i)9-s − 334.·10-s − 271.·11-s + (78.5 + 136. i)12-s + (229. + 132. i)13-s + 715. i·14-s + (965. − 557. i)15-s + (603. + 1.04e3i)16-s + (961. − 554. i)17-s + ⋯
L(s)  = 1  + (0.962 − 0.555i)2-s + (−0.672 + 1.16i)3-s + (0.117 − 0.202i)4-s + (−0.824 − 0.475i)5-s + 1.49i·6-s + (−0.439 + 0.760i)7-s + 0.850i·8-s + (−0.404 − 0.699i)9-s − 1.05·10-s − 0.677·11-s + (0.157 + 0.272i)12-s + (0.376 + 0.217i)13-s + 0.975i·14-s + (1.10 − 0.639i)15-s + (0.589 + 1.02i)16-s + (0.806 − 0.465i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.497 - 0.867i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.497 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $-0.497 - 0.867i$
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: \((2,\ 37,\ (\ :5/2),\ -0.497 - 0.867i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.595247 + 1.02764i\)
\(L(\frac12)\) \(\approx\) \(0.595247 + 1.02764i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (-6.09e3 + 5.67e3i)T \)
good2 \( 1 + (-5.44 + 3.14i)T + (16 - 27.7i)T^{2} \)
3 \( 1 + (10.4 - 18.1i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (46.0 + 26.6i)T + (1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + (56.9 - 98.6i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 + 271.T + 1.61e5T^{2} \)
13 \( 1 + (-229. - 132. i)T + (1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 + (-961. + 554. i)T + (7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-576. - 332. i)T + (1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 - 991. iT - 6.43e6T^{2} \)
29 \( 1 + 2.70e3iT - 2.05e7T^{2} \)
31 \( 1 - 9.34e3iT - 2.86e7T^{2} \)
41 \( 1 + (8.72e3 - 1.51e4i)T + (-5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 - 1.67e4iT - 1.47e8T^{2} \)
47 \( 1 + 8.18e3T + 2.29e8T^{2} \)
53 \( 1 + (1.12e4 + 1.94e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-3.02e4 + 1.74e4i)T + (3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-4.78e4 - 2.76e4i)T + (4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-7.55e3 + 1.30e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + (-1.01e3 + 1.75e3i)T + (-9.02e8 - 1.56e9i)T^{2} \)
73 \( 1 + 7.54e4T + 2.07e9T^{2} \)
79 \( 1 + (1.73e4 + 1.00e4i)T + (1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (-3.38e4 - 5.86e4i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 + (-3.33e4 + 1.92e4i)T + (2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 4.56e4iT - 8.58e9T^{2} \)
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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

−15.91457049399617068038383593344, −14.65021615034432596491349398611, −13.07880105736769078133566665305, −11.99934042952069816677216774788, −11.27365359303593891829776912815, −9.817405643590394068504744481482, −8.232014468179060515034034352872, −5.59101281091913152238655725237, −4.60578744932613501412898826244, −3.27846939406728720513866471204, 0.55663310693845561376554096770, 3.72074870849513995346524285646, 5.63363264234079921188526684957, 6.85780005219609559930302916518, 7.67363652971292954213011155252, 10.28775329116962743365566800128, 11.70558457096928780064112667806, 12.87033660476690624989460596421, 13.54560898145369547835315351780, 14.89179247084102405525637256316

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