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Properties

Label 2-370-185.104-c1-0-12
Degree $2$
Conductor $370$
Sign $0.880 + 0.473i$
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  + (0.939 − 0.342i)2-s + (0.0975 − 0.267i)3-s + (0.766 − 0.642i)4-s + (−2.05 + 0.877i)5-s − 0.285i·6-s + (3.86 − 0.681i)7-s + (0.500 − 0.866i)8-s + (2.23 + 1.87i)9-s + (−1.63 + 1.52i)10-s + (2.37 − 4.10i)11-s + (−0.0975 − 0.267i)12-s + (−3.78 + 3.17i)13-s + (3.39 − 1.96i)14-s + (0.0345 + 0.636i)15-s + (0.173 − 0.984i)16-s + (−0.839 − 0.704i)17-s + ⋯
L(s)  = 1  + (0.664 − 0.241i)2-s + (0.0562 − 0.154i)3-s + (0.383 − 0.321i)4-s + (−0.919 + 0.392i)5-s − 0.116i·6-s + (1.46 − 0.257i)7-s + (0.176 − 0.306i)8-s + (0.745 + 0.625i)9-s + (−0.516 + 0.483i)10-s + (0.715 − 1.23i)11-s + (−0.0281 − 0.0773i)12-s + (−1.05 + 0.881i)13-s + (0.908 − 0.524i)14-s + (0.00891 + 0.164i)15-s + (0.0434 − 0.246i)16-s + (−0.203 − 0.170i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 + 0.473i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.880 + 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.880 + 0.473i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.880 + 0.473i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.99210 - 0.501844i\)
\(L(\frac12)\) \(\approx\) \(1.99210 - 0.501844i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.939 + 0.342i)T \)
5 \( 1 + (2.05 - 0.877i)T \)
37 \( 1 + (-1.98 - 5.74i)T \)
good3 \( 1 + (-0.0975 + 0.267i)T + (-2.29 - 1.92i)T^{2} \)
7 \( 1 + (-3.86 + 0.681i)T + (6.57 - 2.39i)T^{2} \)
11 \( 1 + (-2.37 + 4.10i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.78 - 3.17i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (0.839 + 0.704i)T + (2.95 + 16.7i)T^{2} \)
19 \( 1 + (-2.10 + 5.79i)T + (-14.5 - 12.2i)T^{2} \)
23 \( 1 + (-3.19 - 5.53i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.58 + 2.64i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.56iT - 31T^{2} \)
41 \( 1 + (6.32 - 5.31i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + 9.58T + 43T^{2} \)
47 \( 1 + (7.68 - 4.43i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.04 + 0.360i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (7.40 + 1.30i)T + (55.4 + 20.1i)T^{2} \)
61 \( 1 + (-5.48 - 6.53i)T + (-10.5 + 60.0i)T^{2} \)
67 \( 1 + (-3.23 + 0.571i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (13.2 + 4.81i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 - 2.00iT - 73T^{2} \)
79 \( 1 + (-12.7 + 2.25i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (-0.774 + 0.923i)T + (-14.4 - 81.7i)T^{2} \)
89 \( 1 + (2.09 + 0.370i)T + (83.6 + 30.4i)T^{2} \)
97 \( 1 + (5.07 + 8.78i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.47946125120514608368099100602, −10.93439314292732365233636360135, −9.584133925671141739284868473293, −8.317930654380897440729057876063, −7.42978525046962928994224413683, −6.74417124383855886103237332853, −5.00310182791546874962033382557, −4.47196432178066688985880060842, −3.15207205920705507404779076991, −1.54772987129236463106570483156, 1.74236802713167670483781887603, 3.65180654053782965973519973031, 4.59985830208231805884598013436, 5.21276503462496498599153745367, 6.89341253959009196999593890328, 7.62411348643214163856900511043, 8.434851251678485296738567482020, 9.640422353667409759101317076315, 10.72409396442465183839626885211, 11.86679908554125569613054148076

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