Properties

Label 2-370-185.142-c1-0-0
Degree $2$
Conductor $370$
Sign $0.196 - 0.980i$
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  i·2-s + (−1.29 − 1.29i)3-s − 4-s + (−0.390 + 2.20i)5-s + (−1.29 + 1.29i)6-s + (−2.67 − 2.67i)7-s + i·8-s + 0.348i·9-s + (2.20 + 0.390i)10-s + 1.29i·11-s + (1.29 + 1.29i)12-s + 6.92i·13-s + (−2.67 + 2.67i)14-s + (3.35 − 2.34i)15-s + 16-s − 4.85·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.747 − 0.747i)3-s − 0.5·4-s + (−0.174 + 0.984i)5-s + (−0.528 + 0.528i)6-s + (−1.00 − 1.00i)7-s + 0.353i·8-s + 0.116i·9-s + (0.696 + 0.123i)10-s + 0.390i·11-s + (0.373 + 0.373i)12-s + 1.92i·13-s + (−0.713 + 0.713i)14-s + (0.866 − 0.604i)15-s + 0.250·16-s − 1.17·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.196 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.196 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.196 - 0.980i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (327, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.196 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.188340 + 0.154328i\)
\(L(\frac12)\) \(\approx\) \(0.188340 + 0.154328i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 + (0.390 - 2.20i)T \)
37 \( 1 + (0.770 + 6.03i)T \)
good3 \( 1 + (1.29 + 1.29i)T + 3iT^{2} \)
7 \( 1 + (2.67 + 2.67i)T + 7iT^{2} \)
11 \( 1 - 1.29iT - 11T^{2} \)
13 \( 1 - 6.92iT - 13T^{2} \)
17 \( 1 + 4.85T + 17T^{2} \)
19 \( 1 + (-4.69 - 4.69i)T + 19iT^{2} \)
23 \( 1 + 3.33iT - 23T^{2} \)
29 \( 1 + (3.73 - 3.73i)T - 29iT^{2} \)
31 \( 1 + (0.146 + 0.146i)T + 31iT^{2} \)
41 \( 1 - 6.38iT - 41T^{2} \)
43 \( 1 - 2.51iT - 43T^{2} \)
47 \( 1 + (2.93 + 2.93i)T + 47iT^{2} \)
53 \( 1 + (1.30 - 1.30i)T - 53iT^{2} \)
59 \( 1 + (4.70 + 4.70i)T + 59iT^{2} \)
61 \( 1 + (7.81 + 7.81i)T + 61iT^{2} \)
67 \( 1 + (3.10 - 3.10i)T - 67iT^{2} \)
71 \( 1 + 5.59T + 71T^{2} \)
73 \( 1 + (0.134 + 0.134i)T + 73iT^{2} \)
79 \( 1 + (10.8 + 10.8i)T + 79iT^{2} \)
83 \( 1 + (8.74 - 8.74i)T - 83iT^{2} \)
89 \( 1 + (3.83 - 3.83i)T - 89iT^{2} \)
97 \( 1 - 13.2T + 97T^{2} \)
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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

−11.52649709246017781910453435223, −10.92853053723540080812101311582, −9.934091297374598332483427444167, −9.194403217494559960121611346533, −7.44891803474294456307464505626, −6.81595267622483566580779805180, −6.16875532573059074421908815417, −4.35159732045288803439054011010, −3.40095266931311758373976034537, −1.76772786516786713595633535844, 0.17634696682800565398427978327, 3.11323308663633993505312138952, 4.58866017228319490715324785701, 5.53651240370616795746655242550, 5.88629576372518561914933353709, 7.44298557984077863744098136464, 8.545437938584340951408709262783, 9.297139888398845957515727925279, 10.07904705316709307133486113118, 11.22269931676098206194762723551

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