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Properties

Label 2-765-17.13-c1-0-26
Degree $2$
Conductor $765$
Sign $0.302 + 0.953i$
Analytic cond. $6.10855$
Root an. cond. $2.47154$
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  + 2.31i·2-s − 3.35·4-s + (−0.707 − 0.707i)5-s + (2.10 − 2.10i)7-s − 3.12i·8-s + (1.63 − 1.63i)10-s + (−4.30 + 4.30i)11-s − 5.74·13-s + (4.86 + 4.86i)14-s + 0.528·16-s + (−0.757 − 4.05i)17-s − 1.68i·19-s + (2.36 + 2.36i)20-s + (−9.96 − 9.96i)22-s + (−0.819 + 0.819i)23-s + ⋯
L(s)  = 1  + 1.63i·2-s − 1.67·4-s + (−0.316 − 0.316i)5-s + (0.795 − 0.795i)7-s − 1.10i·8-s + (0.517 − 0.517i)10-s + (−1.29 + 1.29i)11-s − 1.59·13-s + (1.30 + 1.30i)14-s + 0.132·16-s + (−0.183 − 0.982i)17-s − 0.385i·19-s + (0.529 + 0.529i)20-s + (−2.12 − 2.12i)22-s + (−0.170 + 0.170i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 765 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.302 + 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 765 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.302 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(765\)    =    \(3^{2} \cdot 5 \cdot 17\)
Sign: $0.302 + 0.953i$
Analytic conductor: \(6.10855\)
Root analytic conductor: \(2.47154\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{765} (676, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 765,\ (\ :1/2),\ 0.302 + 0.953i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0264804 - 0.0193801i\)
\(L(\frac12)\) \(\approx\) \(0.0264804 - 0.0193801i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (0.707 + 0.707i)T \)
17 \( 1 + (0.757 + 4.05i)T \)
good2 \( 1 - 2.31iT - 2T^{2} \)
7 \( 1 + (-2.10 + 2.10i)T - 7iT^{2} \)
11 \( 1 + (4.30 - 4.30i)T - 11iT^{2} \)
13 \( 1 + 5.74T + 13T^{2} \)
19 \( 1 + 1.68iT - 19T^{2} \)
23 \( 1 + (0.819 - 0.819i)T - 23iT^{2} \)
29 \( 1 + (6.36 + 6.36i)T + 29iT^{2} \)
31 \( 1 + (0.0555 + 0.0555i)T + 31iT^{2} \)
37 \( 1 + (3.59 + 3.59i)T + 37iT^{2} \)
41 \( 1 + (-1.33 + 1.33i)T - 41iT^{2} \)
43 \( 1 - 11.6iT - 43T^{2} \)
47 \( 1 - 5.18T + 47T^{2} \)
53 \( 1 - 10.3iT - 53T^{2} \)
59 \( 1 + 5.32iT - 59T^{2} \)
61 \( 1 + (3.68 - 3.68i)T - 61iT^{2} \)
67 \( 1 - 0.279T + 67T^{2} \)
71 \( 1 + (10.4 + 10.4i)T + 71iT^{2} \)
73 \( 1 + (1.23 + 1.23i)T + 73iT^{2} \)
79 \( 1 + (-0.317 + 0.317i)T - 79iT^{2} \)
83 \( 1 - 3.04iT - 83T^{2} \)
89 \( 1 + 3.61T + 89T^{2} \)
97 \( 1 + (-5.51 - 5.51i)T + 97iT^{2} \)
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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

−9.816997598790321442911396311321, −9.175055562557521449699876770394, −7.76134952218471422845214811069, −7.63978360886357039138502872715, −7.08097186707294576572720634322, −5.64065744255907427496027039367, −4.70507893011477668074522211066, −4.52687300975569848216801247309, −2.36716813743286265739539545945, −0.01547991697170397526407102666, 1.90793435649853110142237905761, 2.73553590667405750092863675096, 3.69256175204743155257441872364, 4.94886484125535739623125884528, 5.64074590208240391407572271036, 7.27063853479190913910190235414, 8.313354127046990051627347195298, 8.856936142195505859865630296236, 10.07333354639549634335467394100, 10.60760778424458919402815603485

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