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Properties

Label 2-765-17.15-c1-0-21
Degree $2$
Conductor $765$
Sign $0.968 + 0.250i$
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  + (0.528 + 0.528i)2-s − 1.44i·4-s + (0.923 − 0.382i)5-s + (2.98 + 1.23i)7-s + (1.81 − 1.81i)8-s + (0.690 + 0.286i)10-s + (−1.04 + 2.52i)11-s − 4.31i·13-s + (0.925 + 2.23i)14-s − 0.956·16-s + (3.47 + 2.22i)17-s + (−0.897 − 0.897i)19-s + (−0.551 − 1.33i)20-s + (−1.88 + 0.782i)22-s + (0.188 − 0.454i)23-s + ⋯
L(s)  = 1  + (0.373 + 0.373i)2-s − 0.720i·4-s + (0.413 − 0.171i)5-s + (1.12 + 0.467i)7-s + (0.643 − 0.643i)8-s + (0.218 + 0.0905i)10-s + (−0.315 + 0.761i)11-s − 1.19i·13-s + (0.247 + 0.596i)14-s − 0.239·16-s + (0.841 + 0.539i)17-s + (−0.205 − 0.205i)19-s + (−0.123 − 0.297i)20-s + (−0.402 + 0.166i)22-s + (0.0392 − 0.0948i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 765 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 765 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(765\)    =    \(3^{2} \cdot 5 \cdot 17\)
Sign: $0.968 + 0.250i$
Analytic conductor: \(6.10855\)
Root analytic conductor: \(2.47154\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{765} (406, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 765,\ (\ :1/2),\ 0.968 + 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.26430 - 0.288129i\)
\(L(\frac12)\) \(\approx\) \(2.26430 - 0.288129i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.923 + 0.382i)T \)
17 \( 1 + (-3.47 - 2.22i)T \)
good2 \( 1 + (-0.528 - 0.528i)T + 2iT^{2} \)
7 \( 1 + (-2.98 - 1.23i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (1.04 - 2.52i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + 4.31iT - 13T^{2} \)
19 \( 1 + (0.897 + 0.897i)T + 19iT^{2} \)
23 \( 1 + (-0.188 + 0.454i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (-0.410 + 0.170i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (2.11 + 5.10i)T + (-21.9 + 21.9i)T^{2} \)
37 \( 1 + (-4.09 - 9.88i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-2.00 - 0.830i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (-1.52 + 1.52i)T - 43iT^{2} \)
47 \( 1 + 8.39iT - 47T^{2} \)
53 \( 1 + (1.28 + 1.28i)T + 53iT^{2} \)
59 \( 1 + (-2.13 + 2.13i)T - 59iT^{2} \)
61 \( 1 + (-11.2 - 4.67i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + 4.21T + 67T^{2} \)
71 \( 1 + (-1.48 - 3.59i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (5.97 - 2.47i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (2.76 - 6.67i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (0.160 + 0.160i)T + 83iT^{2} \)
89 \( 1 - 13.3iT - 89T^{2} \)
97 \( 1 + (13.6 - 5.66i)T + (68.5 - 68.5i)T^{2} \)
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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

−10.21204020403347642885841779890, −9.656101161003797436188306464597, −8.400375683458543031179056067072, −7.75611861428365396709788436367, −6.64817936259834700298557935529, −5.52195113468357637233470853542, −5.24106962363765536685742368151, −4.16837878487711372524845904761, −2.43430639573907192008274617959, −1.25675120111379101877402991689, 1.58085129551483291617665127071, 2.76967072329434806405496849367, 3.92060844259626920071037865615, 4.76304422556583169025679282574, 5.75353501231894380559393283273, 7.07695506982585796680182139561, 7.75842485217213241966795925385, 8.590151366620658066052449945396, 9.481456080825146568345471354045, 10.75786815268813310074216612007

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