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Properties

Label 2-765-17.15-c1-0-8
Degree $2$
Conductor $765$
Sign $0.983 + 0.178i$
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  + (−1.81 − 1.81i)2-s + 4.58i·4-s + (0.923 − 0.382i)5-s + (2.79 + 1.15i)7-s + (4.69 − 4.69i)8-s + (−2.37 − 0.981i)10-s + (−0.472 + 1.14i)11-s + 1.66i·13-s + (−2.96 − 7.16i)14-s − 7.85·16-s + (3.62 + 1.96i)17-s + (−2.43 − 2.43i)19-s + (1.75 + 4.23i)20-s + (2.92 − 1.21i)22-s + (−3.15 + 7.62i)23-s + ⋯
L(s)  = 1  + (−1.28 − 1.28i)2-s + 2.29i·4-s + (0.413 − 0.171i)5-s + (1.05 + 0.436i)7-s + (1.65 − 1.65i)8-s + (−0.749 − 0.310i)10-s + (−0.142 + 0.344i)11-s + 0.462i·13-s + (−0.792 − 1.91i)14-s − 1.96·16-s + (0.879 + 0.475i)17-s + (−0.558 − 0.558i)19-s + (0.392 + 0.947i)20-s + (0.624 − 0.258i)22-s + (−0.658 + 1.58i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 765 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.178i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 765 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 + 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(765\)    =    \(3^{2} \cdot 5 \cdot 17\)
Sign: $0.983 + 0.178i$
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: \((2,\ 765,\ (\ :1/2),\ 0.983 + 0.178i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.852448 - 0.0766356i\)
\(L(\frac12)\) \(\approx\) \(0.852448 - 0.0766356i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.62 - 1.96i)T \)
good2 \( 1 + (1.81 + 1.81i)T + 2iT^{2} \)
7 \( 1 + (-2.79 - 1.15i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (0.472 - 1.14i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 - 1.66iT - 13T^{2} \)
19 \( 1 + (2.43 + 2.43i)T + 19iT^{2} \)
23 \( 1 + (3.15 - 7.62i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (8.41 - 3.48i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (-1.24 - 3.00i)T + (-21.9 + 21.9i)T^{2} \)
37 \( 1 + (-2.13 - 5.15i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-8.63 - 3.57i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (-1.12 + 1.12i)T - 43iT^{2} \)
47 \( 1 - 5.84iT - 47T^{2} \)
53 \( 1 + (-4.27 - 4.27i)T + 53iT^{2} \)
59 \( 1 + (-9.77 + 9.77i)T - 59iT^{2} \)
61 \( 1 + (8.06 + 3.33i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + 4.64T + 67T^{2} \)
71 \( 1 + (4.19 + 10.1i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (-7.10 + 2.94i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (5.33 - 12.8i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (6.07 + 6.07i)T + 83iT^{2} \)
89 \( 1 + 9.05iT - 89T^{2} \)
97 \( 1 + (-11.1 + 4.60i)T + (68.5 - 68.5i)T^{2} \)
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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

−10.24011029497790084456325776855, −9.487257026737401897608738762270, −8.884287028564148487876788255878, −7.994520492640026505908455522295, −7.39573785506061828241298977568, −5.83483324993374646278501560533, −4.63738543956778648384627204881, −3.40739043337723159761070807442, −2.10648426724150530328006927587, −1.42373351139923924751340727972, 0.72315347844451538941900847180, 2.15903602484200964494695616485, 4.23389615074738392596551106406, 5.53406878883729852593141290145, 5.98934174794465369806761984855, 7.19666835031416765980060715542, 7.80870442660980779728770162402, 8.431236854948601060120226213728, 9.320272271552618761560156827134, 10.25600479040066477639060956153

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