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Properties

Label 2-370-185.174-c1-0-2
Degree $2$
Conductor $370$
Sign $-0.0823 - 0.996i$
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  + (−0.866 + 0.5i)2-s + (1.73 + i)3-s + (0.499 − 0.866i)4-s + (−2.12 + 0.686i)5-s − 1.99·6-s + (3.73 + 2.15i)7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (1.5 − 1.65i)10-s − 2.31·11-s + (1.73 − 0.999i)12-s + (3.46 + 2i)13-s − 4.31·14-s + (−4.37 − 0.939i)15-s + (−0.5 − 0.866i)16-s + (−1.14 + 0.658i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.999 + 0.577i)3-s + (0.249 − 0.433i)4-s + (−0.951 + 0.306i)5-s − 0.816·6-s + (1.41 + 0.815i)7-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (0.474 − 0.524i)10-s − 0.698·11-s + (0.499 − 0.288i)12-s + (0.960 + 0.554i)13-s − 1.15·14-s + (−1.12 − 0.242i)15-s + (−0.125 − 0.216i)16-s + (−0.276 + 0.159i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0823 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0823 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.0823 - 0.996i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (359, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.0823 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.881359 + 0.957234i\)
\(L(\frac12)\) \(\approx\) \(0.881359 + 0.957234i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + (2.12 - 0.686i)T \)
37 \( 1 + (2.59 + 5.5i)T \)
good3 \( 1 + (-1.73 - i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (-3.73 - 2.15i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + 2.31T + 11T^{2} \)
13 \( 1 + (-3.46 - 2i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.14 - 0.658i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.15 - 2.00i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 - 3.31T + 29T^{2} \)
31 \( 1 + 4.31T + 31T^{2} \)
41 \( 1 + (-2.81 + 4.87i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 10.9iT - 43T^{2} \)
47 \( 1 + 2.31iT - 47T^{2} \)
53 \( 1 + (-4.01 + 2.31i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.316 + 0.548i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.97 + 6.88i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.20 + 4.15i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-7.15 + 12.3i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 10.6iT - 73T^{2} \)
79 \( 1 + (-4.47 + 7.75i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.74 + 3.31i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.816 + 1.41i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 9.94iT - 97T^{2} \)
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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

−11.29996742890319861930413053486, −10.81612119039438955688314804186, −9.505972510456026972621913076766, −8.582322924097021196151610257274, −8.237596433921648948096233153060, −7.37029609991019822274869972995, −5.87118824649724471821682534163, −4.60623432720673260663510940503, −3.45535367933090608461135603418, −2.01058765060270737050008591613, 1.06230187051445519253862898191, 2.52054931702342127146291312207, 3.84610977660767159853472137906, 4.96860297197689377159560756307, 6.98203899839081182292098151196, 7.78637662576620045601547727201, 8.338762998445065185919224031955, 8.808178727393412074859592229935, 10.56899129675340813696389290024, 10.93267930459770442257016094519

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