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Properties

Label 2-765-85.29-c0-0-1
Degree $2$
Conductor $765$
Sign $0.855 + 0.518i$
Analytic cond. $0.381784$
Root an. cond. $0.617887$
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  + (1.53 − 0.636i)2-s + (1.24 − 1.24i)4-s + (−0.195 + 0.980i)5-s + (0.487 − 1.17i)8-s + (0.324 + 1.63i)10-s − 0.351i·16-s + (−0.555 − 0.831i)17-s + (−0.707 − 1.70i)19-s + (0.980 + 1.46i)20-s + (−0.785 + 1.17i)23-s + (−0.923 − 0.382i)25-s + (−0.216 − 0.324i)31-s + (0.263 + 0.636i)32-s + (−1.38 − 0.923i)34-s + (−2.17 − 2.17i)38-s + ⋯
L(s)  = 1  + (1.53 − 0.636i)2-s + (1.24 − 1.24i)4-s + (−0.195 + 0.980i)5-s + (0.487 − 1.17i)8-s + (0.324 + 1.63i)10-s − 0.351i·16-s + (−0.555 − 0.831i)17-s + (−0.707 − 1.70i)19-s + (0.980 + 1.46i)20-s + (−0.785 + 1.17i)23-s + (−0.923 − 0.382i)25-s + (−0.216 − 0.324i)31-s + (0.263 + 0.636i)32-s + (−1.38 − 0.923i)34-s + (−2.17 − 2.17i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 765 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.855 + 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 765 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.855 + 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(765\)    =    \(3^{2} \cdot 5 \cdot 17\)
Sign: $0.855 + 0.518i$
Analytic conductor: \(0.381784\)
Root analytic conductor: \(0.617887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{765} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 765,\ (\ :0),\ 0.855 + 0.518i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.006303975\)
\(L(\frac12)\) \(\approx\) \(2.006303975\)
\(L(1)\) 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 \( 1 + (0.195 - 0.980i)T \)
17 \( 1 + (0.555 + 0.831i)T \)
good2 \( 1 + (-1.53 + 0.636i)T + (0.707 - 0.707i)T^{2} \)
7 \( 1 + (-0.923 + 0.382i)T^{2} \)
11 \( 1 + (-0.382 - 0.923i)T^{2} \)
13 \( 1 - iT^{2} \)
19 \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \)
23 \( 1 + (0.785 - 1.17i)T + (-0.382 - 0.923i)T^{2} \)
29 \( 1 + (-0.923 - 0.382i)T^{2} \)
31 \( 1 + (0.216 + 0.324i)T + (-0.382 + 0.923i)T^{2} \)
37 \( 1 + (-0.382 + 0.923i)T^{2} \)
41 \( 1 + (0.923 - 0.382i)T^{2} \)
43 \( 1 + (0.707 + 0.707i)T^{2} \)
47 \( 1 + (1.38 - 1.38i)T - iT^{2} \)
53 \( 1 + (-1.02 + 0.425i)T + (0.707 - 0.707i)T^{2} \)
59 \( 1 + (0.707 + 0.707i)T^{2} \)
61 \( 1 + (-0.382 - 1.92i)T + (-0.923 + 0.382i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (0.382 - 0.923i)T^{2} \)
73 \( 1 + (-0.923 - 0.382i)T^{2} \)
79 \( 1 + (-1.08 + 1.63i)T + (-0.382 - 0.923i)T^{2} \)
83 \( 1 + (0.425 + 1.02i)T + (-0.707 + 0.707i)T^{2} \)
89 \( 1 - iT^{2} \)
97 \( 1 + (0.923 + 0.382i)T^{2} \)
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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

−10.85856883986466116291881565210, −9.993287738227406643449674263931, −8.899675819723109921300218069086, −7.50017317429704932780608066154, −6.71889077744830482727051401400, −5.88938585876667942437869281021, −4.83929826215152197503707713374, −3.98110080097249245435114092072, −2.97730045998207573032930133339, −2.18680474008867270906849577472, 2.05337541996567255759698809051, 3.72393917394212103147894060438, 4.24073282378413211963634033945, 5.23077706863798531327037182107, 6.01736798568547657879348212639, 6.77623836596798423136559088596, 8.037793785398181806447513423303, 8.468897823046643079527696841765, 9.792970316960638523406190543241, 10.79933943038697198372837746602

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