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Properties

Label 2-775-155.123-c1-0-17
Degree $2$
Conductor $775$
Sign $-0.643 - 0.765i$
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  + (−1.15 + 1.15i)2-s + (−2.13 + 2.13i)3-s − 0.670i·4-s − 4.93i·6-s + (2.15 − 2.15i)7-s + (−1.53 − 1.53i)8-s − 6.11i·9-s + 1.23i·11-s + (1.43 + 1.43i)12-s + (1.84 − 1.84i)13-s + 4.98i·14-s + 4.89·16-s + (5.22 + 5.22i)17-s + (7.06 + 7.06i)18-s + 0.445i·19-s + ⋯
L(s)  = 1  + (−0.817 + 0.817i)2-s + (−1.23 + 1.23i)3-s − 0.335i·4-s − 2.01i·6-s + (0.814 − 0.814i)7-s + (−0.543 − 0.543i)8-s − 2.03i·9-s + 0.373i·11-s + (0.413 + 0.413i)12-s + (0.512 − 0.512i)13-s + 1.33i·14-s + 1.22·16-s + (1.26 + 1.26i)17-s + (1.66 + 1.66i)18-s + 0.102i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.643 - 0.765i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.643 - 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(775\)    =    \(5^{2} \cdot 31\)
Sign: $-0.643 - 0.765i$
Analytic conductor: \(6.18840\)
Root analytic conductor: \(2.48765\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{775} (743, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 775,\ (\ :1/2),\ -0.643 - 0.765i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.285413 + 0.612548i\)
\(L(\frac12)\) \(\approx\) \(0.285413 + 0.612548i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.805 + 5.50i)T \)
good2 \( 1 + (1.15 - 1.15i)T - 2iT^{2} \)
3 \( 1 + (2.13 - 2.13i)T - 3iT^{2} \)
7 \( 1 + (-2.15 + 2.15i)T - 7iT^{2} \)
11 \( 1 - 1.23iT - 11T^{2} \)
13 \( 1 + (-1.84 + 1.84i)T - 13iT^{2} \)
17 \( 1 + (-5.22 - 5.22i)T + 17iT^{2} \)
19 \( 1 - 0.445iT - 19T^{2} \)
23 \( 1 + (3.08 - 3.08i)T - 23iT^{2} \)
29 \( 1 - 3.52T + 29T^{2} \)
37 \( 1 + (1.14 + 1.14i)T + 37iT^{2} \)
41 \( 1 - 7.27T + 41T^{2} \)
43 \( 1 + (-6.07 + 6.07i)T - 43iT^{2} \)
47 \( 1 + (-4.31 + 4.31i)T - 47iT^{2} \)
53 \( 1 + (2.99 - 2.99i)T - 53iT^{2} \)
59 \( 1 + 0.821iT - 59T^{2} \)
61 \( 1 - 13.3iT - 61T^{2} \)
67 \( 1 + (4.27 - 4.27i)T - 67iT^{2} \)
71 \( 1 - 8.57T + 71T^{2} \)
73 \( 1 + (-4.97 + 4.97i)T - 73iT^{2} \)
79 \( 1 + 13.9T + 79T^{2} \)
83 \( 1 + (7.67 - 7.67i)T - 83iT^{2} \)
89 \( 1 + 0.0894T + 89T^{2} \)
97 \( 1 + (-9.56 + 9.56i)T - 97iT^{2} \)
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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

−10.32195677899404833732984656924, −10.00282486766942367275097344393, −8.946332911535008368699020841358, −7.968805296509333162199765677315, −7.30991178280249758273010646395, −6.00458195959042432025586883122, −5.63533703462387558145765645635, −4.28625167667651703801631312938, −3.68094633050094210734419071836, −0.946982278382232162687867447922, 0.77189709257232157371051524116, 1.69743964584806559481009623014, 2.77804970586221150768389759142, 4.91162822891214724587632518364, 5.67932583425846933936366879496, 6.39604007744892563832851244873, 7.58489405294284573233690790790, 8.299240740071966134615962834297, 9.186909757996345777710301474490, 10.23411387580146503527187211973

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