Properties

Label 2-740-740.683-c0-0-0
Degree $2$
Conductor $740$
Sign $0.505 - 0.862i$
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  + (−0.984 + 0.173i)2-s + (0.939 − 0.342i)4-s + (0.866 + 0.5i)5-s + (−0.866 + 0.5i)8-s + (−0.342 + 0.939i)9-s + (−0.939 − 0.342i)10-s + (0.592 + 1.62i)13-s + (0.766 − 0.642i)16-s + (−1.43 − 0.524i)17-s + (0.173 − 0.984i)18-s + (0.984 + 0.173i)20-s + (0.499 + 0.866i)25-s + (−0.866 − 1.5i)26-s + (0.469 − 1.75i)29-s + (−0.642 + 0.766i)32-s + ⋯
L(s)  = 1  + (−0.984 + 0.173i)2-s + (0.939 − 0.342i)4-s + (0.866 + 0.5i)5-s + (−0.866 + 0.5i)8-s + (−0.342 + 0.939i)9-s + (−0.939 − 0.342i)10-s + (0.592 + 1.62i)13-s + (0.766 − 0.642i)16-s + (−1.43 − 0.524i)17-s + (0.173 − 0.984i)18-s + (0.984 + 0.173i)20-s + (0.499 + 0.866i)25-s + (−0.866 − 1.5i)26-s + (0.469 − 1.75i)29-s + (−0.642 + 0.766i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.505 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.505 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(740\)    =    \(2^{2} \cdot 5 \cdot 37\)
Sign: $0.505 - 0.862i$
Analytic conductor: \(0.369308\)
Root analytic conductor: \(0.607707\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{740} (683, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 740,\ (\ :0),\ 0.505 - 0.862i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6974916599\)
\(L(\frac12)\) \(\approx\) \(0.6974916599\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.984 - 0.173i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
37 \( 1 + (0.342 + 0.939i)T \)
good3 \( 1 + (0.342 - 0.939i)T^{2} \)
7 \( 1 + (-0.984 + 0.173i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.592 - 1.62i)T + (-0.766 + 0.642i)T^{2} \)
17 \( 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2} \)
19 \( 1 + (0.342 - 0.939i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.469 + 1.75i)T + (-0.866 - 0.5i)T^{2} \)
31 \( 1 + iT^{2} \)
41 \( 1 + (-0.439 - 1.20i)T + (-0.766 + 0.642i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.866 - 0.5i)T^{2} \)
53 \( 1 + (-0.515 - 0.0451i)T + (0.984 + 0.173i)T^{2} \)
59 \( 1 + (0.984 + 0.173i)T^{2} \)
61 \( 1 + (0.357 - 0.766i)T + (-0.642 - 0.766i)T^{2} \)
67 \( 1 + (0.984 - 0.173i)T^{2} \)
71 \( 1 + (0.939 + 0.342i)T^{2} \)
73 \( 1 + (-1.36 + 1.36i)T - iT^{2} \)
79 \( 1 + (-0.984 + 0.173i)T^{2} \)
83 \( 1 + (0.642 - 0.766i)T^{2} \)
89 \( 1 + (-0.173 - 0.0151i)T + (0.984 + 0.173i)T^{2} \)
97 \( 1 + (-0.342 + 0.592i)T + (-0.5 - 0.866i)T^{2} \)
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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

−10.76067444941930635427768490880, −9.692845659836932800759351100759, −9.118629811949836707827520449992, −8.305353373127105000772759056998, −7.19501785084613290115878201225, −6.50980164620071267723099364071, −5.73044571392788577620396598360, −4.41767307695046762933185835368, −2.58921980027080127982068932535, −1.89788314147824459590205165907, 1.08779499773764868223892473897, 2.52320938570822896644558510632, 3.64144473583051867047408231064, 5.31024753210860842150392367936, 6.19132483606818077106362419926, 6.91318132309444430171713138528, 8.279667606751972608772341154993, 8.760247007851554394987827014167, 9.455686512221119356226464062486, 10.49781458485541886015581776735

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