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Properties

Label 2-740-740.263-c1-0-19
Degree $2$
Conductor $740$
Sign $-0.186 - 0.982i$
Analytic cond. $5.90892$
Root an. cond. $2.43082$
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  + (−1.15 + 0.811i)2-s + (0.684 − 1.87i)4-s + (−2.17 − 0.519i)5-s + (0.732 + 2.73i)8-s + (−1.02 − 2.81i)9-s + (2.94 − 1.16i)10-s + (−4.90 + 2.28i)13-s + (−3.06 − 2.57i)16-s + (7.06 + 3.29i)17-s + (3.47 + 2.43i)18-s + (−2.46 + 3.73i)20-s + (4.46 + 2.25i)25-s + (3.83 − 6.63i)26-s + (−4.01 + 6.95i)29-s + (5.63 + 0.493i)32-s + ⋯
L(s)  = 1  + (−0.819 + 0.573i)2-s + (0.342 − 0.939i)4-s + (−0.972 − 0.232i)5-s + (0.258 + 0.965i)8-s + (−0.342 − 0.939i)9-s + (0.930 − 0.367i)10-s + (−1.36 + 0.634i)13-s + (−0.766 − 0.642i)16-s + (1.71 + 0.799i)17-s + (0.819 + 0.573i)18-s + (−0.550 + 0.834i)20-s + (0.892 + 0.451i)25-s + (0.751 − 1.30i)26-s + (−0.745 + 1.29i)29-s + (0.996 + 0.0871i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.186 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.186 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(740\)    =    \(2^{2} \cdot 5 \cdot 37\)
Sign: $-0.186 - 0.982i$
Analytic conductor: \(5.90892\)
Root analytic conductor: \(2.43082\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{740} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 740,\ (\ :1/2),\ -0.186 - 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.357107 + 0.431209i\)
\(L(\frac12)\) \(\approx\) \(0.357107 + 0.431209i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.15 - 0.811i)T \)
5 \( 1 + (2.17 + 0.519i)T \)
37 \( 1 + (-5.29 - 2.99i)T \)
good3 \( 1 + (1.02 + 2.81i)T^{2} \)
7 \( 1 + (6.89 + 1.21i)T^{2} \)
11 \( 1 + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.90 - 2.28i)T + (8.35 - 9.95i)T^{2} \)
17 \( 1 + (-7.06 - 3.29i)T + (10.9 + 13.0i)T^{2} \)
19 \( 1 + (17.8 - 6.49i)T^{2} \)
23 \( 1 + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (4.01 - 6.95i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 31T^{2} \)
41 \( 1 + (-11.7 - 4.29i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + 43iT^{2} \)
47 \( 1 + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.26 - 14.5i)T + (-52.1 + 9.20i)T^{2} \)
59 \( 1 + (-10.2 - 58.1i)T^{2} \)
61 \( 1 + (1.81 - 4.97i)T + (-46.7 - 39.2i)T^{2} \)
67 \( 1 + (65.9 + 11.6i)T^{2} \)
71 \( 1 + (66.7 - 24.2i)T^{2} \)
73 \( 1 + (7.02 + 7.02i)T + 73iT^{2} \)
79 \( 1 + (-13.7 + 77.7i)T^{2} \)
83 \( 1 + (-53.3 - 63.5i)T^{2} \)
89 \( 1 + (-2.01 - 1.68i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (3.84 - 14.3i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.44140534360214979312985743932, −9.503770865255050967996005272418, −8.964724818684473679895287205519, −7.85244896606736890829881276354, −7.44123510612343512073777996626, −6.38718058077528971102940477932, −5.41924098150301787805046440913, −4.30612666802313543436407907615, −2.99763707570021274780286869494, −1.16538930715563236393422809886, 0.43586906640162340572224103795, 2.38455296404750770360171888989, 3.23272735198086676590769287701, 4.45366329977292066198395414204, 5.59382477536803560413911283357, 7.21039416244575920491609041994, 7.71766872866178397782613112090, 8.191140182296077336845379696679, 9.521589369707075396459512198916, 10.06011120292737490691496746517

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