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Properties

Label 2-370-185.43-c1-0-16
Degree $2$
Conductor $370$
Sign $-0.148 + 0.988i$
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 + (1.41 − 1.41i)3-s − 4-s + (2.12 − 0.707i)5-s + (−1.41 − 1.41i)6-s + (2.70 − 2.70i)7-s + i·8-s − 1.00i·9-s + (−0.707 − 2.12i)10-s + 3.82i·11-s + (−1.41 + 1.41i)12-s + 2.24i·13-s + (−2.70 − 2.70i)14-s + (1.99 − 4i)15-s + 16-s − 1.82·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.816 − 0.816i)3-s − 0.5·4-s + (0.948 − 0.316i)5-s + (−0.577 − 0.577i)6-s + (1.02 − 1.02i)7-s + 0.353i·8-s − 0.333i·9-s + (−0.223 − 0.670i)10-s + 1.15i·11-s + (−0.408 + 0.408i)12-s + 0.621i·13-s + (−0.723 − 0.723i)14-s + (0.516 − 1.03i)15-s + 0.250·16-s − 0.443·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.148 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.148 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.148 + 0.988i$
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: \((2,\ 370,\ (\ :1/2),\ -0.148 + 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30246 - 1.51295i\)
\(L(\frac12)\) \(\approx\) \(1.30246 - 1.51295i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.12 + 0.707i)T \)
37 \( 1 + (4.94 + 3.53i)T \)
good3 \( 1 + (-1.41 + 1.41i)T - 3iT^{2} \)
7 \( 1 + (-2.70 + 2.70i)T - 7iT^{2} \)
11 \( 1 - 3.82iT - 11T^{2} \)
13 \( 1 - 2.24iT - 13T^{2} \)
17 \( 1 + 1.82T + 17T^{2} \)
19 \( 1 + (5.82 - 5.82i)T - 19iT^{2} \)
23 \( 1 + 2.58iT - 23T^{2} \)
29 \( 1 + (6.70 + 6.70i)T + 29iT^{2} \)
31 \( 1 + (4.12 - 4.12i)T - 31iT^{2} \)
41 \( 1 - 7iT - 41T^{2} \)
43 \( 1 + 7iT - 43T^{2} \)
47 \( 1 + (8.24 - 8.24i)T - 47iT^{2} \)
53 \( 1 + (-2.12 - 2.12i)T + 53iT^{2} \)
59 \( 1 + (1.17 - 1.17i)T - 59iT^{2} \)
61 \( 1 + (-9.29 + 9.29i)T - 61iT^{2} \)
67 \( 1 + (-1.24 - 1.24i)T + 67iT^{2} \)
71 \( 1 + 0.343T + 71T^{2} \)
73 \( 1 + (-6 + 6i)T - 73iT^{2} \)
79 \( 1 + (-9.65 + 9.65i)T - 79iT^{2} \)
83 \( 1 + (0.828 + 0.828i)T + 83iT^{2} \)
89 \( 1 + (-7.41 - 7.41i)T + 89iT^{2} \)
97 \( 1 + 7T + 97T^{2} \)
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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

−11.00711349087738868186720552974, −10.29639787717503943501750387594, −9.331109399244402552816478164437, −8.370755527925986912669570481885, −7.59437514731611160660224904917, −6.53112068758936500721581705491, −4.94017374713547160644424335116, −4.01915241725879251382405912823, −2.02731338382684470744773368679, −1.74954329926926999374904816106, 2.26692685529396963735913351243, 3.51354717052311284685673312934, 5.00421882926259100207422635366, 5.69087031663801276301996249390, 6.83667352161600449688762203691, 8.343609756502396127882474464457, 8.784426565370780819042173112217, 9.448138216042431051835946053939, 10.64079587970441200475170543041, 11.35102488369660251188430635614

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