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Properties

Label 2-772-772.59-c0-0-0
Degree $2$
Conductor $772$
Sign $0.216 + 0.976i$
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 + (0.835 − 0.732i)5-s + (0.382 − 0.923i)8-s − i·9-s + (−1.10 + 0.0726i)10-s + (−0.0726 + 0.108i)13-s + (−0.866 + 0.499i)16-s + (−0.369 + 0.125i)17-s + (−0.608 + 0.793i)18-s + (0.923 + 0.617i)20-s + (0.0306 − 0.232i)25-s + (0.123 − 0.0420i)26-s + (1.65 − 1.10i)29-s + (0.991 + 0.130i)32-s + ⋯
L(s)  = 1  + (−0.793 − 0.608i)2-s + (0.258 + 0.965i)4-s + (0.835 − 0.732i)5-s + (0.382 − 0.923i)8-s − i·9-s + (−1.10 + 0.0726i)10-s + (−0.0726 + 0.108i)13-s + (−0.866 + 0.499i)16-s + (−0.369 + 0.125i)17-s + (−0.608 + 0.793i)18-s + (0.923 + 0.617i)20-s + (0.0306 − 0.232i)25-s + (0.123 − 0.0420i)26-s + (1.65 − 1.10i)29-s + (0.991 + 0.130i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 772 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.216 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 772 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.216 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(772\)    =    \(2^{2} \cdot 193\)
Sign: $0.216 + 0.976i$
Analytic conductor: \(0.385278\)
Root analytic conductor: \(0.620707\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{772} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 772,\ (\ :0),\ 0.216 + 0.976i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7543504618\)
\(L(\frac12)\) \(\approx\) \(0.7543504618\)
\(L(1)\) 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 \( 1 + (-0.835 + 0.732i)T + (0.130 - 0.991i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.923 - 0.382i)T^{2} \)
13 \( 1 + (0.0726 - 0.108i)T + (-0.382 - 0.923i)T^{2} \)
17 \( 1 + (0.369 - 0.125i)T + (0.793 - 0.608i)T^{2} \)
19 \( 1 + (-0.130 + 0.991i)T^{2} \)
23 \( 1 + (-0.707 + 0.707i)T^{2} \)
29 \( 1 + (-1.65 + 1.10i)T + (0.382 - 0.923i)T^{2} \)
31 \( 1 + (-0.258 - 0.965i)T^{2} \)
37 \( 1 + (1.18 - 0.583i)T + (0.608 - 0.793i)T^{2} \)
41 \( 1 + (0.284 + 0.837i)T + (-0.793 + 0.608i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.608 - 0.793i)T^{2} \)
53 \( 1 + (-0.284 - 0.576i)T + (-0.608 + 0.793i)T^{2} \)
59 \( 1 + (-0.866 - 0.5i)T^{2} \)
61 \( 1 + (1.34 + 1.18i)T + (0.130 + 0.991i)T^{2} \)
67 \( 1 + (-0.707 + 0.707i)T^{2} \)
71 \( 1 + (0.923 - 0.382i)T^{2} \)
73 \( 1 + (-1.69 - 0.837i)T + (0.608 + 0.793i)T^{2} \)
79 \( 1 + (0.130 + 0.991i)T^{2} \)
83 \( 1 + (-0.965 - 0.258i)T^{2} \)
89 \( 1 + (-0.923 - 1.38i)T + (-0.382 + 0.923i)T^{2} \)
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

−10.14878440916916296652800947917, −9.465704889116356430995130149955, −8.884438388856590639572000198003, −8.107282463415026287327104989414, −6.88587130965198074075293638437, −6.11555918147251931886390463618, −4.80584177165102345495934889931, −3.68127047933723864205308670972, −2.40978064899841668991239670395, −1.14505044608192245189447664147, 1.79641561234089062297370619511, 2.81878817228431584719077475117, 4.74404726100170277344062439497, 5.57905234363733090158627832331, 6.55134581371877915527208542736, 7.16637485593342775378504731372, 8.189205396183756694542612389538, 8.924990761349674152823677643632, 9.982615193053473833982341854198, 10.46356416608449500561009814548

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