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Properties

Label 2-370-185.103-c1-0-8
Degree $2$
Conductor $370$
Sign $0.238 - 0.971i$
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 + (1.81 − 1.30i)5-s + (−0.658 + 0.658i)6-s + (0.0267 + 0.0999i)7-s − 0.999·8-s + (1.84 − 1.06i)9-s + (2.03 + 0.917i)10-s + 5.56i·11-s + (−0.900 − 0.241i)12-s + (−0.858 + 1.48i)13-s + (−0.0731 + 0.0731i)14-s + (1.61 + 1.31i)15-s + (−0.5 − 0.866i)16-s + (6.18 − 3.57i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.139 + 0.519i)3-s + (−0.249 + 0.433i)4-s + (0.811 − 0.584i)5-s + (−0.269 + 0.269i)6-s + (0.0101 + 0.0377i)7-s − 0.353·8-s + (0.615 − 0.355i)9-s + (0.644 + 0.290i)10-s + 1.67i·11-s + (−0.259 − 0.0696i)12-s + (−0.238 + 0.412i)13-s + (−0.0195 + 0.0195i)14-s + (0.416 + 0.340i)15-s + (−0.125 − 0.216i)16-s + (1.50 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.238 - 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.238 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.238 - 0.971i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.238 - 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49435 + 1.17123i\)
\(L(\frac12)\) \(\approx\) \(1.49435 + 1.17123i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.81 + 1.30i)T \)
37 \( 1 + (-5.72 - 2.05i)T \)
good3 \( 1 + (-0.241 - 0.900i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (-0.0267 - 0.0999i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 - 5.56iT - 11T^{2} \)
13 \( 1 + (0.858 - 1.48i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-6.18 + 3.57i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.98 - 0.530i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + 3.96T + 23T^{2} \)
29 \( 1 + (5.95 - 5.95i)T - 29iT^{2} \)
31 \( 1 + (5.87 + 5.87i)T + 31iT^{2} \)
41 \( 1 + (8.14 + 4.70i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 - 1.18T + 43T^{2} \)
47 \( 1 + (-4.09 + 4.09i)T - 47iT^{2} \)
53 \( 1 + (-0.924 + 3.44i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-1.94 + 7.26i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (6.44 - 1.72i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (0.575 - 0.154i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-3.42 + 5.92i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.83 + 2.83i)T - 73iT^{2} \)
79 \( 1 + (12.0 - 3.22i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (-1.87 + 6.98i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (0.962 + 0.257i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + 16.0iT - 97T^{2} \)
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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

−11.96922480208707289010707612977, −10.25804632762167516784545714101, −9.658332053924175536421757222951, −9.064611105958485736707540842269, −7.64241072647829480093741191857, −6.88848665593988133923660340874, −5.58218331084894904204772774365, −4.78614610041499965805412214864, −3.80832077434190181103475902916, −1.94487673554362970642613309497, 1.42143026937833509379230097607, 2.73773244204452211529133540287, 3.84763613726655378665195099431, 5.57410099318990557575159605310, 6.09913793937222845402041658340, 7.45065938155584162210555569269, 8.397432065274685881045496437574, 9.655469328546103943574552258537, 10.45353095063423259106055740847, 11.05974212910792056961418144907

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