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Properties

Label 2-775-155.92-c1-0-4
Degree $2$
Conductor $775$
Sign $0.846 - 0.532i$
Analytic cond. $6.18840$
Root an. cond. $2.48765$
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  + (−1.22 − 1.22i)2-s + (1 + i)3-s + 0.999i·4-s − 2.44i·6-s + (−2.44 − 2.44i)7-s + (−1.22 + 1.22i)8-s − i·9-s + 4.89i·11-s + (−0.999 + 0.999i)12-s + (3 + 3i)13-s + 5.99i·14-s + 5·16-s + (−5 + 5i)17-s + (−1.22 + 1.22i)18-s + 2i·19-s + ⋯
L(s)  = 1  + (−0.866 − 0.866i)2-s + (0.577 + 0.577i)3-s + 0.499i·4-s − 0.999i·6-s + (−0.925 − 0.925i)7-s + (−0.433 + 0.433i)8-s − 0.333i·9-s + 1.47i·11-s + (−0.288 + 0.288i)12-s + (0.832 + 0.832i)13-s + 1.60i·14-s + 1.25·16-s + (−1.21 + 1.21i)17-s + (−0.288 + 0.288i)18-s + 0.458i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 - 0.532i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.846 - 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(775\)    =    \(5^{2} \cdot 31\)
Sign: $0.846 - 0.532i$
Analytic conductor: \(6.18840\)
Root analytic conductor: \(2.48765\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{775} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 775,\ (\ :1/2),\ 0.846 - 0.532i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.793438 + 0.228882i\)
\(L(\frac12)\) \(\approx\) \(0.793438 + 0.228882i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
31 \( 1 + (-5 - 2.44i)T \)
good2 \( 1 + (1.22 + 1.22i)T + 2iT^{2} \)
3 \( 1 + (-1 - i)T + 3iT^{2} \)
7 \( 1 + (2.44 + 2.44i)T + 7iT^{2} \)
11 \( 1 - 4.89iT - 11T^{2} \)
13 \( 1 + (-3 - 3i)T + 13iT^{2} \)
17 \( 1 + (5 - 5i)T - 17iT^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
23 \( 1 + (1 + i)T + 23iT^{2} \)
29 \( 1 - 9.79T + 29T^{2} \)
37 \( 1 + (3 - 3i)T - 37iT^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + (3 + 3i)T + 43iT^{2} \)
47 \( 1 + (-7.34 - 7.34i)T + 47iT^{2} \)
53 \( 1 + (1 + i)T + 53iT^{2} \)
59 \( 1 - 12iT - 59T^{2} \)
61 \( 1 - 4.89iT - 61T^{2} \)
67 \( 1 + (-2.44 - 2.44i)T + 67iT^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (-3 - 3i)T + 73iT^{2} \)
79 \( 1 + 9.79T + 79T^{2} \)
83 \( 1 + (-7 - 7i)T + 83iT^{2} \)
89 \( 1 + 9.79T + 89T^{2} \)
97 \( 1 + 97iT^{2} \)
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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

−10.14683299405734496587947842652, −9.828216038408659231238059752876, −8.838076090962932887868236933530, −8.381543079324415432202936142024, −6.87447693510428892794130438812, −6.31912055739719243568992698753, −4.44836869538229856589841273497, −3.81555652663731661937354925359, −2.64393752199386394337912722768, −1.36275471905547943828447314752, 0.55122398265623553145506840992, 2.64782530428220073399081818967, 3.31325501974616662085799357108, 5.23721848333884946808318964118, 6.29093879947763659673154942575, 6.73723717497165015025694443247, 7.892626918603343501059025405928, 8.602325863418570863414516053170, 8.849610738611567790211443684632, 9.882059004221634076702975156565

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