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Properties

Label 2-370-185.97-c1-0-9
Degree $2$
Conductor $370$
Sign $0.946 - 0.322i$
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  + (−0.5 + 0.866i)2-s + (−0.241 + 0.900i)3-s + (−0.499 − 0.866i)4-s + (2.17 − 0.519i)5-s + (−0.658 − 0.658i)6-s + (0.596 − 2.22i)7-s + 0.999·8-s + (1.84 + 1.06i)9-s + (−0.637 + 2.14i)10-s − 4.55i·11-s + (0.900 − 0.241i)12-s + (−2.24 − 3.88i)13-s + (1.62 + 1.62i)14-s + (−0.0572 + 2.08i)15-s + (−0.5 + 0.866i)16-s + (0.121 + 0.0699i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.139 + 0.519i)3-s + (−0.249 − 0.433i)4-s + (0.972 − 0.232i)5-s + (−0.269 − 0.269i)6-s + (0.225 − 0.841i)7-s + 0.353·8-s + (0.615 + 0.355i)9-s + (−0.201 + 0.677i)10-s − 1.37i·11-s + (0.259 − 0.0696i)12-s + (−0.622 − 1.07i)13-s + (0.435 + 0.435i)14-s + (−0.0147 + 0.537i)15-s + (−0.125 + 0.216i)16-s + (0.0293 + 0.0169i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 - 0.322i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.946 - 0.322i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.946 - 0.322i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.946 - 0.322i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28191 + 0.212670i\)
\(L(\frac12)\) \(\approx\) \(1.28191 + 0.212670i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (-2.17 + 0.519i)T \)
37 \( 1 + (2.51 - 5.53i)T \)
good3 \( 1 + (0.241 - 0.900i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (-0.596 + 2.22i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + 4.55iT - 11T^{2} \)
13 \( 1 + (2.24 + 3.88i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.121 - 0.0699i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.17 - 1.11i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 - 0.446T + 23T^{2} \)
29 \( 1 + (-4.72 - 4.72i)T + 29iT^{2} \)
31 \( 1 + (4.75 - 4.75i)T - 31iT^{2} \)
41 \( 1 + (-1.14 + 0.659i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + 9.82T + 43T^{2} \)
47 \( 1 + (-4.73 - 4.73i)T + 47iT^{2} \)
53 \( 1 + (0.694 + 2.59i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.78 + 10.3i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-10.7 - 2.88i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (8.53 + 2.28i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (7.49 + 12.9i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.51 - 3.51i)T + 73iT^{2} \)
79 \( 1 + (-8.56 - 2.29i)T + (68.4 + 39.5i)T^{2} \)
83 \( 1 + (-0.647 - 2.41i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (17.1 - 4.58i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 - 11.6iT - 97T^{2} \)
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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

−10.95224385634253901195735672682, −10.37721713794618902336258157906, −9.738956786600275880682425028761, −8.669817493296787716749204127620, −7.72159650729549086687091000952, −6.70948478861976866827801117838, −5.47574017364619384850717320524, −4.91596063157079365099296687593, −3.31039309656927263989363384343, −1.19383863532323832498297540449, 1.69476201171934995485527908276, 2.45678291400891107132686192695, 4.32271886074056810325727401985, 5.50032122260540275780474743813, 6.80181834924241748686430076424, 7.42523630346804766148505342898, 8.935964792673732921134572024579, 9.630988702798691824637141300505, 10.17279741327359662512949360756, 11.61298215238718135634583418485

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