Properties

Label 2-736-736.45-c0-0-2
Degree $2$
Conductor $736$
Sign $-0.442 + 0.896i$
Analytic cond. $0.367311$
Root an. cond. $0.606062$
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  + (−0.965 + 0.258i)2-s + (0.758 − 1.83i)3-s + (0.866 − 0.499i)4-s + (−0.258 + 1.96i)6-s + (−0.707 + 0.707i)8-s + (−2.07 − 2.07i)9-s + (−0.258 − 1.96i)12-s + (−1.12 − 0.465i)13-s + (0.500 − 0.866i)16-s + (2.53 + 1.46i)18-s + (0.707 + 0.707i)23-s + (0.758 + 1.83i)24-s + (0.707 − 0.707i)25-s + (1.20 + 0.158i)26-s + (−3.53 + 1.46i)27-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)2-s + (0.758 − 1.83i)3-s + (0.866 − 0.499i)4-s + (−0.258 + 1.96i)6-s + (−0.707 + 0.707i)8-s + (−2.07 − 2.07i)9-s + (−0.258 − 1.96i)12-s + (−1.12 − 0.465i)13-s + (0.500 − 0.866i)16-s + (2.53 + 1.46i)18-s + (0.707 + 0.707i)23-s + (0.758 + 1.83i)24-s + (0.707 − 0.707i)25-s + (1.20 + 0.158i)26-s + (−3.53 + 1.46i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.442 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.442 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(736\)    =    \(2^{5} \cdot 23\)
Sign: $-0.442 + 0.896i$
Analytic conductor: \(0.367311\)
Root analytic conductor: \(0.606062\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{736} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 736,\ (\ :0),\ -0.442 + 0.896i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7266440441\)
\(L(\frac12)\) \(\approx\) \(0.7266440441\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
23 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (-0.758 + 1.83i)T + (-0.707 - 0.707i)T^{2} \)
5 \( 1 + (-0.707 + 0.707i)T^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + (0.707 - 0.707i)T^{2} \)
13 \( 1 + (1.12 + 0.465i)T + (0.707 + 0.707i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + (-0.707 - 0.707i)T^{2} \)
29 \( 1 + (0.0999 - 0.241i)T + (-0.707 - 0.707i)T^{2} \)
31 \( 1 - 1.93T + T^{2} \)
37 \( 1 + (-0.707 + 0.707i)T^{2} \)
41 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
43 \( 1 + (0.707 - 0.707i)T^{2} \)
47 \( 1 - 1.73iT - T^{2} \)
53 \( 1 + (0.707 - 0.707i)T^{2} \)
59 \( 1 + (-0.707 + 0.292i)T + (0.707 - 0.707i)T^{2} \)
61 \( 1 + (0.707 + 0.707i)T^{2} \)
67 \( 1 + (0.707 + 0.707i)T^{2} \)
71 \( 1 + (-0.366 + 0.366i)T - iT^{2} \)
73 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (-0.707 - 0.707i)T^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 - T^{2} \)
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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

−10.04233816416472546298406674106, −9.181643405615934405844130050665, −8.349436622193528711501146763959, −7.80651538665624126046160822518, −6.99532792873946170272137501593, −6.46310517263232890824643508468, −5.31982258028538765972089478961, −3.04588589309996863613779560125, −2.29516063140238464545112928321, −0.990260620520687279354491514291, 2.40728921814532441785786434573, 3.18155143949620179640051353390, 4.34158685626325762415547735793, 5.19761745154719075807069430006, 6.69299313359419398145688915473, 7.87115721794504831279994022004, 8.610952285184835228276831804808, 9.287963508561079401527311209162, 9.939283772043500403511124021135, 10.48526325280835300020033808838

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