Properties

Label 2-370-185.84-c1-0-8
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 + (2.00 − 1.15i)7-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (1.5 + 1.65i)10-s + 4.31·11-s + (−1.73 − 0.999i)12-s + (−3.46 + 2i)13-s + 2.31·14-s + (−4.37 + 0.939i)15-s + (−0.5 + 0.866i)16-s + (−4.60 − 2.65i)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 + (0.758 − 0.437i)7-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (0.474 + 0.524i)10-s + 1.30·11-s + (−0.499 − 0.288i)12-s + (−0.960 + 0.554i)13-s + 0.619·14-s + (−1.12 + 0.242i)15-s + (−0.125 + 0.216i)16-s + (−1.11 − 0.644i)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: $\chi_{370} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.0823 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24447 + 1.14582i\)
\(L(\frac12)\) \(\approx\) \(1.24447 + 1.14582i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.00 + 1.15i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 - 4.31T + 11T^{2} \)
13 \( 1 + (3.46 - 2i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (4.60 + 2.65i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.15 - 3.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 + 3.31T + 29T^{2} \)
31 \( 1 - 2.31T + 31T^{2} \)
41 \( 1 + (3.81 + 6.61i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 8.94iT - 43T^{2} \)
47 \( 1 - 4.31iT - 47T^{2} \)
53 \( 1 + (-7.47 - 4.31i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.31 + 10.9i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.97 + 10.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.45 + 0.841i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.84 - 6.65i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 2.63iT - 73T^{2} \)
79 \( 1 + (5.47 + 9.48i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.74 - 3.31i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-5.81 + 10.0i)T + (-44.5 - 77.0i)T^{2} \)
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

−11.52537850034187015408210688052, −10.95945443276275621818888162153, −9.864716421650195814941422565956, −9.088244821552297638866288962910, −7.47209426271322226984421287013, −6.64298679679868742657955775327, −5.64530715403442425148328128217, −4.89849127946528970334212524799, −3.90661864910432203478654490584, −1.97308957337219426172340879660, 1.22325277491720704754375292076, 2.49877187507108648673795629662, 4.51497283382260931273053777047, 5.29138227166719596708249517742, 6.27567235695224819592929744217, 6.85995393994479590304409287551, 8.514588824413218247022982482442, 9.475267746424306893921298760088, 10.54263182017105035420949528164, 11.53218156817976270789322870202

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