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Properties

Label 2-740-740.503-c0-0-0
Degree $2$
Conductor $740$
Sign $0.0738 + 0.997i$
Analytic cond. $0.369308$
Root an. cond. $0.607707$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.342 − 0.939i)2-s + (−0.766 − 0.642i)4-s + (0.866 + 0.5i)5-s + (−0.866 + 0.500i)8-s + (−0.642 − 0.766i)9-s + (0.766 − 0.642i)10-s + (1.11 − 1.32i)13-s + (0.173 + 0.984i)16-s + (0.266 − 0.223i)17-s + (−0.939 + 0.342i)18-s + (−0.342 − 0.939i)20-s + (0.499 + 0.866i)25-s + (−0.866 − 1.5i)26-s + (−0.0451 + 0.168i)29-s + (0.984 + 0.173i)32-s + ⋯
L(s)  = 1  + (0.342 − 0.939i)2-s + (−0.766 − 0.642i)4-s + (0.866 + 0.5i)5-s + (−0.866 + 0.500i)8-s + (−0.642 − 0.766i)9-s + (0.766 − 0.642i)10-s + (1.11 − 1.32i)13-s + (0.173 + 0.984i)16-s + (0.266 − 0.223i)17-s + (−0.939 + 0.342i)18-s + (−0.342 − 0.939i)20-s + (0.499 + 0.866i)25-s + (−0.866 − 1.5i)26-s + (−0.0451 + 0.168i)29-s + (0.984 + 0.173i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0738 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0738 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(740\)    =    \(2^{2} \cdot 5 \cdot 37\)
Sign: $0.0738 + 0.997i$
Analytic conductor: \(0.369308\)
Root analytic conductor: \(0.607707\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{740} (503, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 740,\ (\ :0),\ 0.0738 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.158852796\)
\(L(\frac12)\) \(\approx\) \(1.158852796\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.342 + 0.939i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
37 \( 1 + (0.642 - 0.766i)T \)
good3 \( 1 + (0.642 + 0.766i)T^{2} \)
7 \( 1 + (0.342 - 0.939i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-1.11 + 1.32i)T + (-0.173 - 0.984i)T^{2} \)
17 \( 1 + (-0.266 + 0.223i)T + (0.173 - 0.984i)T^{2} \)
19 \( 1 + (0.642 + 0.766i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.0451 - 0.168i)T + (-0.866 - 0.5i)T^{2} \)
31 \( 1 + iT^{2} \)
41 \( 1 + (1.26 - 1.50i)T + (-0.173 - 0.984i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.866 - 0.5i)T^{2} \)
53 \( 1 + (0.296 - 0.424i)T + (-0.342 - 0.939i)T^{2} \)
59 \( 1 + (-0.342 - 0.939i)T^{2} \)
61 \( 1 + (1.98 - 0.173i)T + (0.984 - 0.173i)T^{2} \)
67 \( 1 + (-0.342 + 0.939i)T^{2} \)
71 \( 1 + (-0.766 + 0.642i)T^{2} \)
73 \( 1 + (-1.36 + 1.36i)T - iT^{2} \)
79 \( 1 + (0.342 - 0.939i)T^{2} \)
83 \( 1 + (-0.984 - 0.173i)T^{2} \)
89 \( 1 + (0.939 - 1.34i)T + (-0.342 - 0.939i)T^{2} \)
97 \( 1 + (-0.642 + 1.11i)T + (-0.5 - 0.866i)T^{2} \)
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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

−10.50733088647965605413220662555, −9.709998483739437550595636282015, −8.946592861516240996782505906850, −8.046911896728215331770763806755, −6.44023723227622479248983280244, −5.88896963221914836240500578338, −4.97166116463857824505649402507, −3.41118207752876735468449731671, −2.93857423065377951281780404827, −1.36393930898467415438610800549, 1.95533852789938805872682765108, 3.56630428540586114876625315696, 4.69854631262294566071030599095, 5.54384860460821255306841519865, 6.25161313841460959628998517898, 7.15838186385874494956314729981, 8.397003286804499869779320658519, 8.775764497111639506649229372663, 9.663075851420577240695465007623, 10.73037512874462722989774984860

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