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Properties

Label 2-370-185.43-c1-0-2
Degree $2$
Conductor $370$
Sign $-0.200 - 0.979i$
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 + (−1.78 + 1.78i)3-s − 4-s + (−0.250 − 2.22i)5-s + (1.78 + 1.78i)6-s + (0.501 − 0.501i)7-s + i·8-s − 3.36i·9-s + (−2.22 + 0.250i)10-s + 5.77i·11-s + (1.78 − 1.78i)12-s − 0.334i·13-s + (−0.501 − 0.501i)14-s + (4.41 + 3.51i)15-s + 16-s − 4.86·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−1.02 + 1.02i)3-s − 0.5·4-s + (−0.112 − 0.993i)5-s + (0.728 + 0.728i)6-s + (0.189 − 0.189i)7-s + 0.353i·8-s − 1.12i·9-s + (−0.702 + 0.0793i)10-s + 1.74i·11-s + (0.514 − 0.514i)12-s − 0.0926i·13-s + (−0.134 − 0.134i)14-s + (1.13 + 0.907i)15-s + 0.250·16-s − 1.17·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.200 - 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.200 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.200 - 0.979i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.200 - 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.287427 + 0.352328i\)
\(L(\frac12)\) \(\approx\) \(0.287427 + 0.352328i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.250 + 2.22i)T \)
37 \( 1 + (0.361 - 6.07i)T \)
good3 \( 1 + (1.78 - 1.78i)T - 3iT^{2} \)
7 \( 1 + (-0.501 + 0.501i)T - 7iT^{2} \)
11 \( 1 - 5.77iT - 11T^{2} \)
13 \( 1 + 0.334iT - 13T^{2} \)
17 \( 1 + 4.86T + 17T^{2} \)
19 \( 1 + (5.60 - 5.60i)T - 19iT^{2} \)
23 \( 1 - 1.33iT - 23T^{2} \)
29 \( 1 + (-7.32 - 7.32i)T + 29iT^{2} \)
31 \( 1 + (2.92 - 2.92i)T - 31iT^{2} \)
41 \( 1 + 7.31iT - 41T^{2} \)
43 \( 1 - 1.15iT - 43T^{2} \)
47 \( 1 + (-4.83 + 4.83i)T - 47iT^{2} \)
53 \( 1 + (6.77 + 6.77i)T + 53iT^{2} \)
59 \( 1 + (-8.73 + 8.73i)T - 59iT^{2} \)
61 \( 1 + (-0.472 + 0.472i)T - 61iT^{2} \)
67 \( 1 + (1.79 + 1.79i)T + 67iT^{2} \)
71 \( 1 + 3.33T + 71T^{2} \)
73 \( 1 + (4.96 - 4.96i)T - 73iT^{2} \)
79 \( 1 + (5.53 - 5.53i)T - 79iT^{2} \)
83 \( 1 + (-4.77 - 4.77i)T + 83iT^{2} \)
89 \( 1 + (3.08 + 3.08i)T + 89iT^{2} \)
97 \( 1 + 3.56T + 97T^{2} \)
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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

−11.62756261359950364046886967511, −10.60027664352078660325619204131, −10.12828687656213828873173151550, −9.204006053214821310328371776782, −8.257392487987734138393126611876, −6.74197925062793969094786826089, −5.33208815880952575174769433523, −4.61484580298512404086381875942, −4.01946354736888056965640771582, −1.79251765398061431086694660031, 0.34322603763008956912382371808, 2.56903374375007360020823423796, 4.34554262776826793018126092069, 5.85814822405197788849412699431, 6.32911528430373170688223342841, 7.03216970924473374430266616848, 8.107442099415404081430015249013, 8.988969021347615247046868077558, 10.64395194753837812319402492921, 11.17674134080015533906571939408

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