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Properties

Label 2-37-37.7-c1-0-1
Degree $2$
Conductor $37$
Sign $0.936 + 0.349i$
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  + (1.26 − 0.460i)2-s + (−1.76 − 0.642i)3-s + (−0.141 + 0.118i)4-s + (0.233 + 1.32i)5-s − 2.53·6-s + (−0.120 − 0.684i)7-s + (−1.47 + 2.54i)8-s + (0.407 + 0.342i)9-s + (0.907 + 1.57i)10-s + (3.20 − 5.55i)11-s + (0.326 − 0.118i)12-s + (−1.75 + 1.47i)13-s + (−0.467 − 0.810i)14-s + (0.439 − 2.49i)15-s + (−0.624 + 3.54i)16-s + (2.97 + 2.49i)17-s + ⋯
L(s)  = 1  + (0.895 − 0.325i)2-s + (−1.01 − 0.371i)3-s + (−0.0707 + 0.0593i)4-s + (0.104 + 0.593i)5-s − 1.03·6-s + (−0.0455 − 0.258i)7-s + (−0.520 + 0.901i)8-s + (0.135 + 0.114i)9-s + (0.287 + 0.497i)10-s + (0.966 − 1.67i)11-s + (0.0942 − 0.0342i)12-s + (−0.486 + 0.408i)13-s + (−0.125 − 0.216i)14-s + (0.113 − 0.643i)15-s + (−0.156 + 0.885i)16-s + (0.720 + 0.604i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 + 0.349i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.936 + 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(37\)
Sign: $0.936 + 0.349i$
Analytic conductor: \(0.295446\)
Root analytic conductor: \(0.543549\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 37,\ (\ :1/2),\ 0.936 + 0.349i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.803896 - 0.145182i\)
\(L(\frac12)\) \(\approx\) \(0.803896 - 0.145182i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 + (6.07 - 0.300i)T \)
good2 \( 1 + (-1.26 + 0.460i)T + (1.53 - 1.28i)T^{2} \)
3 \( 1 + (1.76 + 0.642i)T + (2.29 + 1.92i)T^{2} \)
5 \( 1 + (-0.233 - 1.32i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (0.120 + 0.684i)T + (-6.57 + 2.39i)T^{2} \)
11 \( 1 + (-3.20 + 5.55i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.75 - 1.47i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-2.97 - 2.49i)T + (2.95 + 16.7i)T^{2} \)
19 \( 1 + (5.08 + 1.85i)T + (14.5 + 12.2i)T^{2} \)
23 \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.979 + 1.69i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 7.10T + 31T^{2} \)
41 \( 1 + (-0.549 + 0.460i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + 2.65T + 43T^{2} \)
47 \( 1 + (0.134 + 0.232i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.926 + 5.25i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (-0.358 + 2.03i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (1.67 - 1.40i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-0.169 - 0.962i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (-6.80 - 2.47i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 - 7.17T + 73T^{2} \)
79 \( 1 + (-2.04 - 11.5i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (10.7 + 9.01i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (0.486 - 2.75i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (1.26 + 2.18i)T + (-48.5 + 84.0i)T^{2} \)
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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

−16.78142597035349169256616560223, −14.77322162823205778574379150474, −13.92064580271603059252311481345, −12.71162883299319140198500785446, −11.61687921144728778248352265881, −10.89587636486628131137131972024, −8.769994558300804100130486581015, −6.66334338680751707543511185307, −5.52609627865233976331304823098, −3.56211514741737062481516002541, 4.49120572591766572743500284337, 5.36246073652903943922138103173, 6.79889965805827076259024102173, 9.177997270010244256287755882724, 10.40304630688820584920917355952, 12.23235680294066707057605478879, 12.61500631001761154893090371433, 14.39224111154898057434983581909, 15.16962797529154630202034309106, 16.54390761203251155403604095012

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