Properties

Label 2-475-475.113-c1-0-30
Degree $2$
Conductor $475$
Sign $0.965 + 0.259i$
Analytic cond. $3.79289$
Root an. cond. $1.94753$
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  + (1.90 − 0.618i)4-s + (0.876 + 2.05i)5-s + (2.71 − 2.71i)7-s + (−1.76 − 2.42i)9-s + (0.475 + 0.345i)11-s + (3.23 − 2.35i)16-s + (0.320 + 0.628i)17-s + (−4.14 − 1.34i)19-s + (2.93 + 3.37i)20-s + (0.831 + 5.24i)23-s + (−3.46 + 3.60i)25-s + (3.48 − 6.83i)28-s + (7.95 + 3.20i)35-s + (−4.85 − 3.52i)36-s + (3.56 + 3.56i)43-s + (1.11 + 0.363i)44-s + ⋯
L(s)  = 1  + (0.951 − 0.309i)4-s + (0.392 + 0.919i)5-s + (1.02 − 1.02i)7-s + (−0.587 − 0.809i)9-s + (0.143 + 0.104i)11-s + (0.809 − 0.587i)16-s + (0.0777 + 0.152i)17-s + (−0.951 − 0.309i)19-s + (0.657 + 0.753i)20-s + (0.173 + 1.09i)23-s + (−0.692 + 0.721i)25-s + (0.658 − 1.29i)28-s + (1.34 + 0.541i)35-s + (−0.809 − 0.587i)36-s + (0.544 + 0.544i)43-s + (0.168 + 0.0547i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.259i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 + 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.965 + 0.259i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.965 + 0.259i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.93514 - 0.255486i\)
\(L(\frac12)\) \(\approx\) \(1.93514 - 0.255486i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.876 - 2.05i)T \)
19 \( 1 + (4.14 + 1.34i)T \)
good2 \( 1 + (-1.90 + 0.618i)T^{2} \)
3 \( 1 + (1.76 + 2.42i)T^{2} \)
7 \( 1 + (-2.71 + 2.71i)T - 7iT^{2} \)
11 \( 1 + (-0.475 - 0.345i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-12.3 - 4.01i)T^{2} \)
17 \( 1 + (-0.320 - 0.628i)T + (-9.99 + 13.7i)T^{2} \)
23 \( 1 + (-0.831 - 5.24i)T + (-21.8 + 7.10i)T^{2} \)
29 \( 1 + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (25.0 + 18.2i)T^{2} \)
37 \( 1 + (35.1 + 11.4i)T^{2} \)
41 \( 1 + (-12.6 + 38.9i)T^{2} \)
43 \( 1 + (-3.56 - 3.56i)T + 43iT^{2} \)
47 \( 1 + (-5.44 - 2.77i)T + (27.6 + 38.0i)T^{2} \)
53 \( 1 + (31.1 + 42.8i)T^{2} \)
59 \( 1 + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (12.6 + 9.17i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (39.3 - 54.2i)T^{2} \)
71 \( 1 + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (-2.66 - 16.8i)T + (-69.4 + 22.5i)T^{2} \)
79 \( 1 + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (16.2 - 8.26i)T + (48.7 - 67.1i)T^{2} \)
89 \( 1 + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-57.0 - 78.4i)T^{2} \)
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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

−11.09764737134394001193316426554, −10.32214976477412627352176678031, −9.374733302167585095691957292999, −8.012388509128245191542052014087, −7.17590339841672927156909727177, −6.45112381966452739110641181284, −5.52653390287293869482559305215, −4.00759291302712743954481317738, −2.79145219949649193266104568770, −1.46564947011107402042837090040, 1.79733667205269090022989020817, 2.60246239406096631291636439046, 4.46145907876378445083672414247, 5.46721390638750128737748665073, 6.18346827110364213572309960220, 7.61080715056042416107318768607, 8.450850507967608390944951145686, 8.882113214729705157371976873435, 10.39545663389310127525738493746, 11.10053436233860991012096616071

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