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Properties

Label 2-775-155.59-c1-0-1
Degree $2$
Conductor $775$
Sign $0.928 - 0.370i$
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  + (−1.48 − 2.03i)2-s + (−1.93 − 0.203i)3-s + (−1.34 + 4.13i)4-s + (2.44 + 4.23i)6-s + (−0.173 − 0.816i)7-s + (5.61 − 1.82i)8-s + (0.756 + 0.160i)9-s + (−3.03 − 3.37i)11-s + (3.43 − 7.70i)12-s + (−2.56 − 5.75i)13-s + (−1.40 + 1.56i)14-s + (−4.99 − 3.63i)16-s + (−0.961 − 0.865i)17-s + (−0.792 − 1.77i)18-s + (−0.526 − 0.234i)19-s + ⋯
L(s)  = 1  + (−1.04 − 1.44i)2-s + (−1.11 − 0.117i)3-s + (−0.671 + 2.06i)4-s + (0.998 + 1.72i)6-s + (−0.0656 − 0.308i)7-s + (1.98 − 0.644i)8-s + (0.252 + 0.0536i)9-s + (−0.915 − 1.01i)11-s + (0.990 − 2.22i)12-s + (−0.710 − 1.59i)13-s + (−0.376 + 0.417i)14-s + (−1.24 − 0.907i)16-s + (−0.233 − 0.209i)17-s + (−0.186 − 0.419i)18-s + (−0.120 − 0.0537i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 - 0.370i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.928 - 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(775\)    =    \(5^{2} \cdot 31\)
Sign: $0.928 - 0.370i$
Analytic conductor: \(6.18840\)
Root analytic conductor: \(2.48765\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{775} (524, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 775,\ (\ :1/2),\ 0.928 - 0.370i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0244238 + 0.00469648i\)
\(L(\frac12)\) \(\approx\) \(0.0244238 + 0.00469648i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.56 - 3.18i)T \)
good2 \( 1 + (1.48 + 2.03i)T + (-0.618 + 1.90i)T^{2} \)
3 \( 1 + (1.93 + 0.203i)T + (2.93 + 0.623i)T^{2} \)
7 \( 1 + (0.173 + 0.816i)T + (-6.39 + 2.84i)T^{2} \)
11 \( 1 + (3.03 + 3.37i)T + (-1.14 + 10.9i)T^{2} \)
13 \( 1 + (2.56 + 5.75i)T + (-8.69 + 9.66i)T^{2} \)
17 \( 1 + (0.961 + 0.865i)T + (1.77 + 16.9i)T^{2} \)
19 \( 1 + (0.526 + 0.234i)T + (12.7 + 14.1i)T^{2} \)
23 \( 1 + (2.24 - 0.729i)T + (18.6 - 13.5i)T^{2} \)
29 \( 1 + (-0.640 + 0.465i)T + (8.96 - 27.5i)T^{2} \)
37 \( 1 + (9.77 - 5.64i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.17 + 11.1i)T + (-40.1 + 8.52i)T^{2} \)
43 \( 1 + (4.54 - 10.2i)T + (-28.7 - 31.9i)T^{2} \)
47 \( 1 + (-4.89 + 6.73i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (1.10 - 5.22i)T + (-48.4 - 21.5i)T^{2} \)
59 \( 1 + (0.557 - 5.30i)T + (-57.7 - 12.2i)T^{2} \)
61 \( 1 + 4.18T + 61T^{2} \)
67 \( 1 + (-6.10 - 3.52i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.94 - 0.626i)T + (64.8 + 28.8i)T^{2} \)
73 \( 1 + (7.69 - 6.92i)T + (7.63 - 72.6i)T^{2} \)
79 \( 1 + (0.856 - 0.950i)T + (-8.25 - 78.5i)T^{2} \)
83 \( 1 + (-13.0 + 1.37i)T + (81.1 - 17.2i)T^{2} \)
89 \( 1 + (1.14 - 3.51i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-5.15 - 1.67i)T + (78.4 + 57.0i)T^{2} \)
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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

−10.46900934340280131632531770853, −10.04510715098110433207738432043, −8.710406675836978484727455399221, −8.136315204776111235355507288167, −7.14306311819927763030842236264, −5.79944783287247114140951649680, −4.96924860301681395221701779772, −3.42584040662771417675301777901, −2.59708138956009922851431597394, −0.864383337786003935134934044126, 0.02814893270827586946907848699, 2.03812216996521542619847103531, 4.55736906336608077022285427860, 5.15435215262858007711412813546, 6.12163372596108508227221949558, 6.77179009856839878008371612045, 7.50966512412649588894610793489, 8.478034735472379879347435369479, 9.347035642480848873644188143948, 10.06497347445943452053198561778

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