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Properties

Label 2-370-185.184-c1-0-14
Degree $2$
Conductor $370$
Sign $0.0524 + 0.998i$
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  − 2-s − 1.78i·3-s + 4-s + (2.21 + 0.288i)5-s + 1.78i·6-s − 3.14i·7-s − 8-s − 0.191·9-s + (−2.21 − 0.288i)10-s − 0.908·11-s − 1.78i·12-s + 2.22·13-s + 3.14i·14-s + (0.515 − 3.96i)15-s + 16-s + 2.10·17-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.03i·3-s + 0.5·4-s + (0.991 + 0.129i)5-s + 0.729i·6-s − 1.19i·7-s − 0.353·8-s − 0.0638·9-s + (−0.701 − 0.0912i)10-s − 0.274·11-s − 0.515i·12-s + 0.616·13-s + 0.841i·14-s + (0.133 − 1.02i)15-s + 0.250·16-s + 0.509·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0524 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0524 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.0524 + 0.998i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.0524 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.848063 - 0.804688i\)
\(L(\frac12)\) \(\approx\) \(0.848063 - 0.804688i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 + (-2.21 - 0.288i)T \)
37 \( 1 + (1.10 + 5.98i)T \)
good3 \( 1 + 1.78iT - 3T^{2} \)
7 \( 1 + 3.14iT - 7T^{2} \)
11 \( 1 + 0.908T + 11T^{2} \)
13 \( 1 - 2.22T + 13T^{2} \)
17 \( 1 - 2.10T + 17T^{2} \)
19 \( 1 - 4.16iT - 19T^{2} \)
23 \( 1 + 7.66T + 23T^{2} \)
29 \( 1 + 2.69iT - 29T^{2} \)
31 \( 1 + 5.96iT - 31T^{2} \)
41 \( 1 - 2.32T + 41T^{2} \)
43 \( 1 + 5.72T + 43T^{2} \)
47 \( 1 - 8.89iT - 47T^{2} \)
53 \( 1 - 9.37iT - 53T^{2} \)
59 \( 1 - 5.55iT - 59T^{2} \)
61 \( 1 + 3.16iT - 61T^{2} \)
67 \( 1 + 7.64iT - 67T^{2} \)
71 \( 1 - 13.5T + 71T^{2} \)
73 \( 1 - 2.14iT - 73T^{2} \)
79 \( 1 - 3.35iT - 79T^{2} \)
83 \( 1 - 16.2iT - 83T^{2} \)
89 \( 1 - 8.35iT - 89T^{2} \)
97 \( 1 + 5.14T + 97T^{2} \)
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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

−10.93311478416110175623544204684, −10.18706826507796261457118577573, −9.528979062470287457169428344913, −8.058520561651285588831565421431, −7.57977442408283067483058896221, −6.49842131276060442174703403325, −5.83079593375521467014380628849, −3.93717686973128500184503330128, −2.18381892287677649437099199069, −1.09669338376445786986416008131, 1.90476899800056057084679979235, 3.25735311500383785121737984036, 4.93214466796485927075300162484, 5.73501832478646686255905080357, 6.78747363313445153171335262825, 8.390049586890769144566011106766, 8.954913639755865302826710882537, 9.873319237429144766601952566215, 10.29952972534424313571894397360, 11.37497910416606447769736510039

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