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Properties

Label 2-775-155.39-c1-0-27
Degree $2$
Conductor $775$
Sign $-0.998 + 0.0504i$
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  + (−2.35 − 0.766i)2-s + (−1.47 + 0.478i)3-s + (3.36 + 2.44i)4-s + 3.84·6-s + (2.32 − 3.20i)7-s + (−3.14 − 4.32i)8-s + (−0.483 + 0.351i)9-s + (0.0194 + 0.0141i)11-s + (−6.12 − 1.99i)12-s + (−4.71 + 1.53i)13-s + (−7.94 + 5.77i)14-s + (1.53 + 4.71i)16-s + (0.358 + 0.493i)17-s + (1.41 − 0.458i)18-s + (2.17 − 6.69i)19-s + ⋯
L(s)  = 1  + (−1.66 − 0.542i)2-s + (−0.851 + 0.276i)3-s + (1.68 + 1.22i)4-s + 1.56·6-s + (0.879 − 1.20i)7-s + (−1.11 − 1.52i)8-s + (−0.161 + 0.117i)9-s + (0.00587 + 0.00426i)11-s + (−1.76 − 0.574i)12-s + (−1.30 + 0.424i)13-s + (−2.12 + 1.54i)14-s + (0.382 + 1.17i)16-s + (0.0870 + 0.119i)17-s + (0.332 − 0.108i)18-s + (0.499 − 1.53i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0504i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0504i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(775\)    =    \(5^{2} \cdot 31\)
Sign: $-0.998 + 0.0504i$
Analytic conductor: \(6.18840\)
Root analytic conductor: \(2.48765\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{775} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 775,\ (\ :1/2),\ -0.998 + 0.0504i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00400314 - 0.158479i\)
\(L(\frac12)\) \(\approx\) \(0.00400314 - 0.158479i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.899 + 5.49i)T \)
good2 \( 1 + (2.35 + 0.766i)T + (1.61 + 1.17i)T^{2} \)
3 \( 1 + (1.47 - 0.478i)T + (2.42 - 1.76i)T^{2} \)
7 \( 1 + (-2.32 + 3.20i)T + (-2.16 - 6.65i)T^{2} \)
11 \( 1 + (-0.0194 - 0.0141i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (4.71 - 1.53i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.358 - 0.493i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2.17 + 6.69i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-3.37 - 4.65i)T + (-7.10 + 21.8i)T^{2} \)
29 \( 1 + (2.12 - 6.54i)T + (-23.4 - 17.0i)T^{2} \)
37 \( 1 + 0.922iT - 37T^{2} \)
41 \( 1 + (-1.08 + 3.32i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + (-3.18 - 1.03i)T + (34.7 + 25.2i)T^{2} \)
47 \( 1 + (7.58 - 2.46i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (5.79 + 7.97i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.701 - 2.15i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + 7.34T + 61T^{2} \)
67 \( 1 - 6.55iT - 67T^{2} \)
71 \( 1 + (-0.670 + 0.486i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.135 + 0.187i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (12.0 - 8.73i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (8.26 + 2.68i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (10.8 + 7.86i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-7.83 + 10.7i)T + (-29.9 - 92.2i)T^{2} \)
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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

−9.901431311980972068473584770208, −9.312629661849198855149431523675, −8.264550127981942853673764087579, −7.31483928390453199270931257199, −7.02575502214355330444368369173, −5.33057084878167818526085014852, −4.47733694320177284943313610756, −2.84501318782671103254564816642, −1.46791661973562444773812912203, −0.16579327345394279850070769008, 1.41032692393444835621668158781, 2.65258473621395471785542454306, 4.98456601113627804386370580619, 5.73414465711907843642185116566, 6.46952386699230302760978040686, 7.54510304745463744309183744990, 8.140682528906529278525789061073, 8.958889145120708485339271749343, 9.742081826618520378808858252949, 10.57030532157977953516653108499

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