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Properties

Label 2-370-37.12-c1-0-13
Degree $2$
Conductor $370$
Sign $-0.999 - 0.0311i$
Analytic cond. $2.95446$
Root an. cond. $1.71885$
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  + (0.766 − 0.642i)2-s + (−1.72 − 1.44i)3-s + (0.173 − 0.984i)4-s + (0.939 − 0.342i)5-s − 2.25·6-s + (−2.62 + 0.955i)7-s + (−0.500 − 0.866i)8-s + (0.359 + 2.03i)9-s + (0.5 − 0.866i)10-s + (−2.60 − 4.51i)11-s + (−1.72 + 1.44i)12-s + (0.188 − 1.06i)13-s + (−1.39 + 2.41i)14-s + (−2.11 − 0.770i)15-s + (−0.939 − 0.342i)16-s + (0.998 + 5.66i)17-s + ⋯
L(s)  = 1  + (0.541 − 0.454i)2-s + (−0.995 − 0.835i)3-s + (0.0868 − 0.492i)4-s + (0.420 − 0.152i)5-s − 0.919·6-s + (−0.992 + 0.361i)7-s + (−0.176 − 0.306i)8-s + (0.119 + 0.679i)9-s + (0.158 − 0.273i)10-s + (−0.785 − 1.36i)11-s + (−0.497 + 0.417i)12-s + (0.0521 − 0.295i)13-s + (−0.373 + 0.646i)14-s + (−0.546 − 0.198i)15-s + (−0.234 − 0.0855i)16-s + (0.242 + 1.37i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0311i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0311i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.999 - 0.0311i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.999 - 0.0311i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0128118 + 0.822229i\)
\(L(\frac12)\) \(\approx\) \(0.0128118 + 0.822229i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.766 + 0.642i)T \)
5 \( 1 + (-0.939 + 0.342i)T \)
37 \( 1 + (1.26 + 5.94i)T \)
good3 \( 1 + (1.72 + 1.44i)T + (0.520 + 2.95i)T^{2} \)
7 \( 1 + (2.62 - 0.955i)T + (5.36 - 4.49i)T^{2} \)
11 \( 1 + (2.60 + 4.51i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.188 + 1.06i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-0.998 - 5.66i)T + (-15.9 + 5.81i)T^{2} \)
19 \( 1 + (4.80 + 4.02i)T + (3.29 + 18.7i)T^{2} \)
23 \( 1 + (1.85 - 3.21i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.90 + 3.29i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 8.95T + 31T^{2} \)
41 \( 1 + (-1.51 + 8.57i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 - 5.14T + 43T^{2} \)
47 \( 1 + (-5.27 + 9.14i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.95 + 1.43i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (5.03 + 1.83i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-1.24 + 7.05i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-5.20 + 1.89i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (-4.29 - 3.60i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 - 8.94T + 73T^{2} \)
79 \( 1 + (2.45 - 0.892i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (0.512 + 2.90i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (-12.2 - 4.44i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (2.06 - 3.58i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.98136201161806205366280264433, −10.45715999537097970448046661008, −9.211285025006936711109231440121, −8.084840578088222698421375013630, −6.59221957090612663056923973787, −6.00558280782906990547006808879, −5.40448934154413088413236929655, −3.67837029724740188055952261320, −2.31290158229002855829509150129, −0.49862405794805812481376317927, 2.70764984410423153088471639513, 4.28607749424949731048210883488, 4.92942972072419738762193447880, 6.07305198330845375533776178992, 6.75301650614797473007968480643, 7.889224305222526614824696028133, 9.515004422154104432024594643908, 10.07091242073503834751809799084, 10.80334662908257424618307719277, 12.00421766646971870006272160971

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