Properties

Label 2-772-772.543-c0-0-0
Degree $2$
Conductor $772$
Sign $0.200 + 0.979i$
Analytic cond. $0.385278$
Root an. cond. $0.620707$
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  + (0.793 − 0.608i)2-s + (0.258 − 0.965i)4-s + (1.09 − 1.25i)5-s + (−0.382 − 0.923i)8-s + i·9-s + (0.108 − 1.65i)10-s + (−1.65 + 1.10i)13-s + (−0.866 − 0.499i)16-s + (−0.630 + 1.85i)17-s + (0.608 + 0.793i)18-s + (−0.923 − 1.38i)20-s + (−0.230 − 1.75i)25-s + (−0.641 + 1.88i)26-s + (0.0726 − 0.108i)29-s + (−0.991 + 0.130i)32-s + ⋯
L(s)  = 1  + (0.793 − 0.608i)2-s + (0.258 − 0.965i)4-s + (1.09 − 1.25i)5-s + (−0.382 − 0.923i)8-s + i·9-s + (0.108 − 1.65i)10-s + (−1.65 + 1.10i)13-s + (−0.866 − 0.499i)16-s + (−0.630 + 1.85i)17-s + (0.608 + 0.793i)18-s + (−0.923 − 1.38i)20-s + (−0.230 − 1.75i)25-s + (−0.641 + 1.88i)26-s + (0.0726 − 0.108i)29-s + (−0.991 + 0.130i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 772 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.200 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 772 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.200 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(772\)    =    \(2^{2} \cdot 193\)
Sign: $0.200 + 0.979i$
Analytic conductor: \(0.385278\)
Root analytic conductor: \(0.620707\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{772} (543, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 772,\ (\ :0),\ 0.200 + 0.979i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.593504751\)
\(L(\frac12)\) \(\approx\) \(1.593504751\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.793 + 0.608i)T \)
193 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 - iT^{2} \)
5 \( 1 + (-1.09 + 1.25i)T + (-0.130 - 0.991i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.923 - 0.382i)T^{2} \)
13 \( 1 + (1.65 - 1.10i)T + (0.382 - 0.923i)T^{2} \)
17 \( 1 + (0.630 - 1.85i)T + (-0.793 - 0.608i)T^{2} \)
19 \( 1 + (0.130 + 0.991i)T^{2} \)
23 \( 1 + (-0.707 - 0.707i)T^{2} \)
29 \( 1 + (-0.0726 + 0.108i)T + (-0.382 - 0.923i)T^{2} \)
31 \( 1 + (-0.258 + 0.965i)T^{2} \)
37 \( 1 + (-0.665 + 1.34i)T + (-0.608 - 0.793i)T^{2} \)
41 \( 1 + (-1.69 - 0.576i)T + (0.793 + 0.608i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.608 - 0.793i)T^{2} \)
53 \( 1 + (1.69 + 0.837i)T + (0.608 + 0.793i)T^{2} \)
59 \( 1 + (-0.866 + 0.5i)T^{2} \)
61 \( 1 + (0.583 + 0.665i)T + (-0.130 + 0.991i)T^{2} \)
67 \( 1 + (-0.707 - 0.707i)T^{2} \)
71 \( 1 + (-0.923 - 0.382i)T^{2} \)
73 \( 1 + (0.284 + 0.576i)T + (-0.608 + 0.793i)T^{2} \)
79 \( 1 + (-0.130 + 0.991i)T^{2} \)
83 \( 1 + (-0.965 + 0.258i)T^{2} \)
89 \( 1 + (0.923 + 0.617i)T + (0.382 + 0.923i)T^{2} \)
97 \( 1 + (-1.53 - 1.17i)T + (0.258 + 0.965i)T^{2} \)
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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

−10.35988008475861393692140986167, −9.585158963662942052419639848669, −8.984581166205286639715306222716, −7.78147166818829515352485031956, −6.47817104017422390799876499399, −5.65150804036874722717112497249, −4.79233894184359297722234156361, −4.24681793743350476998382924556, −2.30372572472568799329027068713, −1.77458625032616306547956977416, 2.69835023649434470594164621772, 2.90060114328725907837314534759, 4.55230118676507945507072376667, 5.52643131041539395159799079946, 6.32132991212048498568004614349, 7.08999391787933954292588984834, 7.65399826092376402001980978009, 9.214449413593543848326105520610, 9.734468538704966937908439017936, 10.74752353809727545996135505119

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