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Properties

Label 2-370-185.23-c1-0-11
Degree $2$
Conductor $370$
Sign $0.718 + 0.695i$
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 + (−2.83 − 0.758i)3-s + (0.499 + 0.866i)4-s + (2.17 − 0.537i)5-s + (−2.07 − 2.07i)6-s + (−1.69 − 0.455i)7-s + 0.999i·8-s + (4.84 + 2.79i)9-s + (2.14 + 0.619i)10-s − 5.29i·11-s + (−0.758 − 2.83i)12-s + (0.342 − 0.197i)13-s + (−1.24 − 1.24i)14-s + (−6.55 − 0.123i)15-s + (−0.5 + 0.866i)16-s + (1.85 − 3.21i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−1.63 − 0.438i)3-s + (0.249 + 0.433i)4-s + (0.970 − 0.240i)5-s + (−0.846 − 0.846i)6-s + (−0.641 − 0.172i)7-s + 0.353i·8-s + (1.61 + 0.932i)9-s + (0.679 + 0.195i)10-s − 1.59i·11-s + (−0.219 − 0.817i)12-s + (0.0951 − 0.0549i)13-s + (−0.332 − 0.332i)14-s + (−1.69 − 0.0319i)15-s + (−0.125 + 0.216i)16-s + (0.449 − 0.778i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.718 + 0.695i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.718 + 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.718 + 0.695i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.718 + 0.695i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13425 - 0.459435i\)
\(L(\frac12)\) \(\approx\) \(1.13425 - 0.459435i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.17 + 0.537i)T \)
37 \( 1 + (5.53 + 2.51i)T \)
good3 \( 1 + (2.83 + 0.758i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (1.69 + 0.455i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + 5.29iT - 11T^{2} \)
13 \( 1 + (-0.342 + 0.197i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.85 + 3.21i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.24 - 1.40i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + 7.28iT - 23T^{2} \)
29 \( 1 + (-1.33 - 1.33i)T + 29iT^{2} \)
31 \( 1 + (-3.75 + 3.75i)T - 31iT^{2} \)
41 \( 1 + (5.24 - 3.03i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 - 8.65iT - 43T^{2} \)
47 \( 1 + (7.10 - 7.10i)T - 47iT^{2} \)
53 \( 1 + (-8.69 + 2.33i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-1.49 - 5.59i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.793 + 0.212i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (0.917 - 3.42i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (1.22 + 2.12i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (7.76 - 7.76i)T - 73iT^{2} \)
79 \( 1 + (-12.4 - 3.33i)T + (68.4 + 39.5i)T^{2} \)
83 \( 1 + (-1.93 + 0.519i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (0.441 - 0.118i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 - 10.7T + 97T^{2} \)
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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

−11.48286181050920815593418384840, −10.58460512747870438243178970612, −9.705256625523794479851563737159, −8.320692721728217128285454127231, −6.95196196996013125551885960991, −6.21877657435212065092975922273, −5.65783601528812457545358630329, −4.80959964109266689195477228418, −3.05599650868621959120543261039, −0.905345188162775184270438573799, 1.65952504020883978525880156223, 3.49983536988256510929001913319, 4.95158639822852505736848024599, 5.46298740940602859242707553410, 6.45699916161448333644970227093, 7.14338061617368971890382834975, 9.375628539006320866031649881944, 10.09494038282766921221761698924, 10.44459688167742526408044539516, 11.75366073879816106908694783528

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