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Properties

Label 2-370-185.84-c1-0-5
Degree $2$
Conductor $370$
Sign $0.346 - 0.937i$
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  + (−0.866 − 0.5i)2-s + (−2.59 + 1.5i)3-s + (0.499 + 0.866i)4-s + (1.86 + 1.23i)5-s + 3·6-s + (1.73 − i)7-s − 0.999i·8-s + (3 − 5.19i)9-s + (−1 − 2i)10-s + (−2.59 − 1.50i)12-s + (−0.866 + 0.5i)13-s − 1.99·14-s + (−6.69 − 0.401i)15-s + (−0.5 + 0.866i)16-s + (5.19 + 3i)17-s + (−5.19 + 3i)18-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−1.49 + 0.866i)3-s + (0.249 + 0.433i)4-s + (0.834 + 0.550i)5-s + 1.22·6-s + (0.654 − 0.377i)7-s − 0.353i·8-s + (1 − 1.73i)9-s + (−0.316 − 0.632i)10-s + (−0.749 − 0.433i)12-s + (−0.240 + 0.138i)13-s − 0.534·14-s + (−1.72 − 0.103i)15-s + (−0.125 + 0.216i)16-s + (1.26 + 0.727i)17-s + (−1.22 + 0.707i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.346 - 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.346 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $0.346 - 0.937i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 0.346 - 0.937i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.604099 + 0.420707i\)
\(L(\frac12)\) \(\approx\) \(0.604099 + 0.420707i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (-1.86 - 1.23i)T \)
37 \( 1 + (2.59 - 5.5i)T \)
good3 \( 1 + (2.59 - 1.5i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (-1.73 + i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + (0.866 - 0.5i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-5.19 - 3i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 7T + 31T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 11iT - 43T^{2} \)
47 \( 1 + 10iT - 47T^{2} \)
53 \( 1 + (4.33 + 2.5i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3 - 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3 - 5.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.46 + 2i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6 - 10.3i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 16iT - 73T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-10.3 - 6i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-9 + 15.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 2iT - 97T^{2} \)
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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

−11.34896799693750865279841531781, −10.49769813531943912216914294285, −10.12960930537706640478830071520, −9.274251884688801881117442933997, −7.80575625953731455632041101998, −6.65884709314959055669187457443, −5.73875777345416712412209856575, −4.81231169690432551583639904574, −3.45091684749219197023854511969, −1.42671085587742827106994379323, 0.832509754432644538921762940639, 2.06851461827463446720219309276, 5.00015311891654345776487521748, 5.46434089816479261096177866478, 6.38270827380729445985997300648, 7.33074003837997448596123214736, 8.279395367813752248781138509478, 9.428761391688619250493208166098, 10.42139052048577421645045276470, 11.16460102767434582980410268330

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