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Properties

Label 2-37-37.3-c1-0-2
Degree $2$
Conductor $37$
Sign $0.721 + 0.692i$
Analytic cond. $0.295446$
Root an. cond. $0.543549$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.48 − 1.77i)2-s + (−2.36 + 1.98i)3-s + (−0.585 − 3.32i)4-s + (−0.253 + 0.696i)5-s + 7.15i·6-s + (−1.02 − 0.373i)7-s + (−2.75 − 1.58i)8-s + (1.13 − 6.41i)9-s + (0.858 + 1.48i)10-s + (−0.376 + 0.651i)11-s + (7.96 + 6.68i)12-s + (2.68 − 0.472i)13-s + (−2.19 + 1.26i)14-s + (−0.782 − 2.14i)15-s + (−0.586 + 0.213i)16-s + (−5.77 − 1.01i)17-s + ⋯
L(s)  = 1  + (1.05 − 1.25i)2-s + (−1.36 + 1.14i)3-s + (−0.292 − 1.66i)4-s + (−0.113 + 0.311i)5-s + 2.91i·6-s + (−0.388 − 0.141i)7-s + (−0.973 − 0.561i)8-s + (0.377 − 2.13i)9-s + (0.271 + 0.470i)10-s + (−0.113 + 0.196i)11-s + (2.30 + 1.93i)12-s + (0.743 − 0.131i)13-s + (−0.586 + 0.338i)14-s + (−0.201 − 0.554i)15-s + (−0.146 + 0.0533i)16-s + (−1.40 − 0.247i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.721 + 0.692i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.721 + 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.721 + 0.692i$
Analytic conductor: \(0.295446\)
Root analytic conductor: \(0.543549\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :1/2),\ 0.721 + 0.692i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.773449 - 0.311352i\)
\(L(\frac12)\) \(\approx\) \(0.773449 - 0.311352i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (0.224 - 6.07i)T \)
good2 \( 1 + (-1.48 + 1.77i)T + (-0.347 - 1.96i)T^{2} \)
3 \( 1 + (2.36 - 1.98i)T + (0.520 - 2.95i)T^{2} \)
5 \( 1 + (0.253 - 0.696i)T + (-3.83 - 3.21i)T^{2} \)
7 \( 1 + (1.02 + 0.373i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (0.376 - 0.651i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.68 + 0.472i)T + (12.2 - 4.44i)T^{2} \)
17 \( 1 + (5.77 + 1.01i)T + (15.9 + 5.81i)T^{2} \)
19 \( 1 + (-1.47 - 1.75i)T + (-3.29 + 18.7i)T^{2} \)
23 \( 1 + (3.96 - 2.29i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.18 - 1.83i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.48iT - 31T^{2} \)
41 \( 1 + (0.929 + 5.27i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + 3.73iT - 43T^{2} \)
47 \( 1 + (-2.26 - 3.91i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (11.8 - 4.33i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (1.26 + 3.48i)T + (-45.1 + 37.9i)T^{2} \)
61 \( 1 + (2.14 - 0.378i)T + (57.3 - 20.8i)T^{2} \)
67 \( 1 + (-4.49 - 1.63i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-2.02 + 1.69i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 - 10.3T + 73T^{2} \)
79 \( 1 + (-2.84 + 7.80i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (-0.488 + 2.77i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (-4.96 - 13.6i)T + (-68.1 + 57.2i)T^{2} \)
97 \( 1 + (-0.861 + 0.497i)T + (48.5 - 84.0i)T^{2} \)
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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

−16.05734013410283849834033949912, −15.18204215899467971166486359973, −13.64700230229218723914466858809, −12.37052175753133555992614686589, −11.32220021480129665835581897232, −10.69444977905674560003087960958, −9.632111816225522775860906252949, −6.24879925168430773268026007002, −4.88774580614743777362381093074, −3.65646125861576805955934975133, 4.74655846318169567945592336148, 6.12866928915493983104911318246, 6.77215989279496157814366133766, 8.251888870440711251234983925193, 11.00281253916290118313931333075, 12.34288132240599741844350075402, 13.04761963305510953796900582387, 13.98403636739718407927280433941, 15.80104284086235224362893380055, 16.32518373073747834536146287227

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