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Properties

Label 2-772-772.751-c0-0-0
Degree $2$
Conductor $772$
Sign $0.882 - 0.470i$
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  + (−0.608 − 0.793i)2-s + (−0.258 + 0.965i)4-s + (0.0255 + 0.389i)5-s + (0.923 − 0.382i)8-s + i·9-s + (0.293 − 0.257i)10-s + (−0.257 + 1.29i)13-s + (−0.866 − 0.499i)16-s + (−1.49 + 0.735i)17-s + (0.793 − 0.608i)18-s + (−0.382 − 0.0761i)20-s + (0.840 − 0.110i)25-s + (1.18 − 0.583i)26-s + (1.47 − 0.293i)29-s + (0.130 + 0.991i)32-s + ⋯
L(s)  = 1  + (−0.608 − 0.793i)2-s + (−0.258 + 0.965i)4-s + (0.0255 + 0.389i)5-s + (0.923 − 0.382i)8-s + i·9-s + (0.293 − 0.257i)10-s + (−0.257 + 1.29i)13-s + (−0.866 − 0.499i)16-s + (−1.49 + 0.735i)17-s + (0.793 − 0.608i)18-s + (−0.382 − 0.0761i)20-s + (0.840 − 0.110i)25-s + (1.18 − 0.583i)26-s + (1.47 − 0.293i)29-s + (0.130 + 0.991i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 772 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 - 0.470i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 772 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 - 0.470i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(772\)    =    \(2^{2} \cdot 193\)
Sign: $0.882 - 0.470i$
Analytic conductor: \(0.385278\)
Root analytic conductor: \(0.620707\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{772} (751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 772,\ (\ :0),\ 0.882 - 0.470i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6448584698\)
\(L(\frac12)\) \(\approx\) \(0.6448584698\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.608 + 0.793i)T \)
193 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 - iT^{2} \)
5 \( 1 + (-0.0255 - 0.389i)T + (-0.991 + 0.130i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.382 + 0.923i)T^{2} \)
13 \( 1 + (0.257 - 1.29i)T + (-0.923 - 0.382i)T^{2} \)
17 \( 1 + (1.49 - 0.735i)T + (0.608 - 0.793i)T^{2} \)
19 \( 1 + (0.991 - 0.130i)T^{2} \)
23 \( 1 + (0.707 + 0.707i)T^{2} \)
29 \( 1 + (-1.47 + 0.293i)T + (0.923 - 0.382i)T^{2} \)
31 \( 1 + (0.258 - 0.965i)T^{2} \)
37 \( 1 + (-0.641 - 1.88i)T + (-0.793 + 0.608i)T^{2} \)
41 \( 1 + (0.837 + 1.69i)T + (-0.608 + 0.793i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.793 + 0.608i)T^{2} \)
53 \( 1 + (-0.837 + 0.284i)T + (0.793 - 0.608i)T^{2} \)
59 \( 1 + (-0.866 + 0.5i)T^{2} \)
61 \( 1 + (-0.0420 + 0.641i)T + (-0.991 - 0.130i)T^{2} \)
67 \( 1 + (0.707 + 0.707i)T^{2} \)
71 \( 1 + (-0.382 + 0.923i)T^{2} \)
73 \( 1 + (0.576 - 1.69i)T + (-0.793 - 0.608i)T^{2} \)
79 \( 1 + (-0.991 - 0.130i)T^{2} \)
83 \( 1 + (0.965 - 0.258i)T^{2} \)
89 \( 1 + (0.382 + 1.92i)T + (-0.923 + 0.382i)T^{2} \)
97 \( 1 + (-1.17 + 1.53i)T + (-0.258 - 0.965i)T^{2} \)
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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

−10.50336692877511135975068422312, −9.997537601447480986907754503126, −8.764173212646871349968390290948, −8.415189211570587710606254892926, −7.15545999153209176938446465253, −6.56213855095717891276412162080, −4.85798124571605869698422225865, −4.13036763470304364881749422575, −2.70431095472434333458226947391, −1.84974517055905239695531485583, 0.864641721165017708832117213964, 2.74704202351608846512647275955, 4.36062392736753711517688895122, 5.23308078999985902458041926396, 6.24749580599410212334550400927, 6.97274959216701284763913480342, 7.930224494978655727659281306033, 8.839835869885333253420026212134, 9.314888727968553574602154571163, 10.29574359487310335162794693288

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