Properties

Label 2-370-185.142-c1-0-2
Degree $2$
Conductor $370$
Sign $0.944 + 0.328i$
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 + (−1.82 − 1.82i)3-s − 4-s + (1.86 + 1.23i)5-s + (−1.82 + 1.82i)6-s + (3.41 + 3.41i)7-s + i·8-s + 3.68i·9-s + (1.23 − 1.86i)10-s + 3.97i·11-s + (1.82 + 1.82i)12-s + 4.28i·13-s + (3.41 − 3.41i)14-s + (−1.14 − 5.66i)15-s + 16-s + 2.57·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−1.05 − 1.05i)3-s − 0.5·4-s + (0.833 + 0.553i)5-s + (−0.746 + 0.746i)6-s + (1.29 + 1.29i)7-s + 0.353i·8-s + 1.22i·9-s + (0.391 − 0.589i)10-s + 1.19i·11-s + (0.527 + 0.527i)12-s + 1.18i·13-s + (0.912 − 0.912i)14-s + (−0.295 − 1.46i)15-s + 0.250·16-s + 0.624·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.328i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 + 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.944 + 0.328i$
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: \((2,\ 370,\ (\ :1/2),\ 0.944 + 0.328i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10449 - 0.186791i\)
\(L(\frac12)\) \(\approx\) \(1.10449 - 0.186791i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.86 - 1.23i)T \)
37 \( 1 + (-2.37 - 5.60i)T \)
good3 \( 1 + (1.82 + 1.82i)T + 3iT^{2} \)
7 \( 1 + (-3.41 - 3.41i)T + 7iT^{2} \)
11 \( 1 - 3.97iT - 11T^{2} \)
13 \( 1 - 4.28iT - 13T^{2} \)
17 \( 1 - 2.57T + 17T^{2} \)
19 \( 1 + (5.24 + 5.24i)T + 19iT^{2} \)
23 \( 1 - 2.21iT - 23T^{2} \)
29 \( 1 + (-2.03 + 2.03i)T - 29iT^{2} \)
31 \( 1 + (3.23 + 3.23i)T + 31iT^{2} \)
41 \( 1 + 2.50iT - 41T^{2} \)
43 \( 1 + 9.00iT - 43T^{2} \)
47 \( 1 + (0.943 + 0.943i)T + 47iT^{2} \)
53 \( 1 + (-2.84 + 2.84i)T - 53iT^{2} \)
59 \( 1 + (-2.91 - 2.91i)T + 59iT^{2} \)
61 \( 1 + (5.71 + 5.71i)T + 61iT^{2} \)
67 \( 1 + (-1.28 + 1.28i)T - 67iT^{2} \)
71 \( 1 - 8.40T + 71T^{2} \)
73 \( 1 + (-5.32 - 5.32i)T + 73iT^{2} \)
79 \( 1 + (9.77 + 9.77i)T + 79iT^{2} \)
83 \( 1 + (-2.57 + 2.57i)T - 83iT^{2} \)
89 \( 1 + (-9.42 + 9.42i)T - 89iT^{2} \)
97 \( 1 - 5.92T + 97T^{2} \)
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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

−11.52970632653148474401320764371, −10.81322621200267962284527506433, −9.626652802848981841799100556451, −8.713993075802850773657328966467, −7.39732220718669784168156898637, −6.48152029514396385066842555934, −5.49763503113679919526937126316, −4.66133348289720941461287967883, −2.22920124009962490139407743789, −1.78620947290343957983582991398, 0.951422231544588587023417947894, 3.84244867733019577258949183559, 4.80803626115700025303694058426, 5.52798455849932445593515210161, 6.26243525237302041343052167426, 7.85131770389014994188847878029, 8.519082583491325524468386169288, 9.860316837825276668151887813998, 10.62483070664788279878711312712, 10.94520852855707102358208702629

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