Properties

Label 2-370-185.104-c1-0-4
Degree $2$
Conductor $370$
Sign $-0.978 - 0.207i$
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 + (−0.691 + 1.90i)3-s + (0.766 − 0.642i)4-s + (1.64 + 1.51i)5-s − 2.02i·6-s + (−2.30 + 0.405i)7-s + (−0.500 + 0.866i)8-s + (−0.835 − 0.701i)9-s + (−2.06 − 0.858i)10-s + (−2.82 + 4.88i)11-s + (0.691 + 1.90i)12-s + (3.98 − 3.34i)13-s + (2.02 − 1.16i)14-s + (−4.01 + 2.08i)15-s + (0.173 − 0.984i)16-s + (−0.926 − 0.777i)17-s + ⋯
L(s)  = 1  + (−0.664 + 0.241i)2-s + (−0.399 + 1.09i)3-s + (0.383 − 0.321i)4-s + (0.736 + 0.676i)5-s − 0.825i·6-s + (−0.870 + 0.153i)7-s + (−0.176 + 0.306i)8-s + (−0.278 − 0.233i)9-s + (−0.652 − 0.271i)10-s + (−0.850 + 1.47i)11-s + (0.199 + 0.548i)12-s + (1.10 − 0.928i)13-s + (0.541 − 0.312i)14-s + (−1.03 + 0.537i)15-s + (0.0434 − 0.246i)16-s + (−0.224 − 0.188i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.207i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 - 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.978 - 0.207i$
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: \((2,\ 370,\ (\ :1/2),\ -0.978 - 0.207i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0747886 + 0.712764i\)
\(L(\frac12)\) \(\approx\) \(0.0747886 + 0.712764i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.64 - 1.51i)T \)
37 \( 1 + (5.96 - 1.16i)T \)
good3 \( 1 + (0.691 - 1.90i)T + (-2.29 - 1.92i)T^{2} \)
7 \( 1 + (2.30 - 0.405i)T + (6.57 - 2.39i)T^{2} \)
11 \( 1 + (2.82 - 4.88i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.98 + 3.34i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (0.926 + 0.777i)T + (2.95 + 16.7i)T^{2} \)
19 \( 1 + (1.34 - 3.69i)T + (-14.5 - 12.2i)T^{2} \)
23 \( 1 + (0.958 + 1.65i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.455 + 0.263i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 9.66iT - 31T^{2} \)
41 \( 1 + (7.68 - 6.44i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 - 0.375T + 43T^{2} \)
47 \( 1 + (-0.377 + 0.217i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-12.1 - 2.13i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (-14.5 - 2.55i)T + (55.4 + 20.1i)T^{2} \)
61 \( 1 + (2.21 + 2.64i)T + (-10.5 + 60.0i)T^{2} \)
67 \( 1 + (-9.61 + 1.69i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (-3.37 - 1.22i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 - 13.4iT - 73T^{2} \)
79 \( 1 + (-7.01 + 1.23i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (5.25 - 6.26i)T + (-14.4 - 81.7i)T^{2} \)
89 \( 1 + (-5.45 - 0.961i)T + (83.6 + 30.4i)T^{2} \)
97 \( 1 + (0.929 + 1.60i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.39038018489308536053441125853, −10.36913754118098057111292310923, −10.12655707758944326309416077094, −9.499947301459844150448402706032, −8.200294800155448219066139628678, −7.03217642764315566269902914536, −6.03852315025471441665287320543, −5.22270339886848881745390801418, −3.73056342888438075007120627757, −2.29051318507899674355476658397, 0.61353406337608710814589099044, 1.92976233429560614597919957011, 3.50087433063965841211337485490, 5.45175049133930683988445343238, 6.39464990183819979775737179156, 6.98829064919784396370182656474, 8.508066426799152610336479072600, 8.847945544498735953076468915821, 10.10549539426860329241722491639, 10.96032350336228225131380943923

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