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Properties

Label 2-370-185.174-c1-0-15
Degree $2$
Conductor $370$
Sign $-0.638 + 0.769i$
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 + (0.499 − 0.866i)4-s + (−2 − i)5-s + (−2.36 − 1.36i)7-s − 0.999i·8-s + (−1.5 − 2.59i)9-s + (−2.23 + 0.133i)10-s + 1.26·11-s + (−1.26 − 0.732i)13-s − 2.73·14-s + (−0.5 − 0.866i)16-s + (1.5 − 0.866i)17-s + (−2.59 − 1.5i)18-s + (0.633 − 1.09i)19-s + (−1.86 + 1.23i)20-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.894 − 0.447i)5-s + (−0.894 − 0.516i)7-s − 0.353i·8-s + (−0.5 − 0.866i)9-s + (−0.705 + 0.0423i)10-s + 0.382·11-s + (−0.351 − 0.203i)13-s − 0.730·14-s + (−0.125 − 0.216i)16-s + (0.363 − 0.210i)17-s + (−0.612 − 0.353i)18-s + (0.145 − 0.251i)19-s + (−0.417 + 0.275i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.638 + 0.769i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.638 + 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $-0.638 + 0.769i$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (359, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ -0.638 + 0.769i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.517103 - 1.10160i\)
\(L(\frac12)\) \(\approx\) \(0.517103 - 1.10160i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 + i)T \)
37 \( 1 + (2.59 + 5.5i)T \)
good3 \( 1 + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (2.36 + 1.36i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 - 1.26T + 11T^{2} \)
13 \( 1 + (1.26 + 0.732i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.5 + 0.866i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.633 + 1.09i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 1.46iT - 23T^{2} \)
29 \( 1 - 1.73T + 29T^{2} \)
31 \( 1 - 7.66T + 31T^{2} \)
41 \( 1 + (2.96 - 5.13i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 8.73iT - 43T^{2} \)
47 \( 1 + 6.73iT - 47T^{2} \)
53 \( 1 + (-7.26 + 4.19i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.73 + 8.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.86 - 6.69i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.56 - 4.36i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.56 + 7.90i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 11.8iT - 73T^{2} \)
79 \( 1 + (-7.83 + 13.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.19 + 1.26i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.23 + 5.59i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 7.19iT - 97T^{2} \)
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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

−11.38417576275058243754349274233, −10.18723161874863083491419109587, −9.375668870956121790978339365576, −8.308309749908091895568614838229, −7.07576806772942669766100840473, −6.26068618965867028918035359577, −4.93002327979365677985497953661, −3.83202000874439967995615623278, −3.03095238161201520402357817312, −0.67490343496289274759006744380, 2.62579009013886535276520687033, 3.60865885480631062549094738059, 4.84222687169843797407332435833, 5.99725054978614868554371862263, 6.90788807005502459156982043082, 7.86809572767764544746799004482, 8.725042785455388759233587068673, 10.00494984877032376205680746778, 10.98700198039297081957049664812, 12.00430646189485048539626074040

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