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Properties

Label 2-475-95.94-c2-0-51
Degree $2$
Conductor $475$
Sign $-0.581 + 0.813i$
Analytic cond. $12.9428$
Root an. cond. $3.59761$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.84·2-s − 2.18·3-s + 4.10·4-s − 6.21·6-s − 9.15i·7-s + 0.287·8-s − 4.22·9-s + 9.89·11-s − 8.95·12-s − 12.3·13-s − 26.0i·14-s − 15.5·16-s − 11.1i·17-s − 12.0·18-s + (−18.7 − 2.97i)19-s + ⋯
L(s)  = 1  + 1.42·2-s − 0.728·3-s + 1.02·4-s − 1.03·6-s − 1.30i·7-s + 0.0359·8-s − 0.469·9-s + 0.899·11-s − 0.746·12-s − 0.948·13-s − 1.86i·14-s − 0.974·16-s − 0.658i·17-s − 0.668·18-s + (−0.987 − 0.156i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.581 + 0.813i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.581 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.581 + 0.813i$
Analytic conductor: \(12.9428\)
Root analytic conductor: \(3.59761\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (474, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1),\ -0.581 + 0.813i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.636610646\)
\(L(\frac12)\) \(\approx\) \(1.636610646\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 + (18.7 + 2.97i)T \)
good2 \( 1 - 2.84T + 4T^{2} \)
3 \( 1 + 2.18T + 9T^{2} \)
7 \( 1 + 9.15iT - 49T^{2} \)
11 \( 1 - 9.89T + 121T^{2} \)
13 \( 1 + 12.3T + 169T^{2} \)
17 \( 1 + 11.1iT - 289T^{2} \)
23 \( 1 + 20.8iT - 529T^{2} \)
29 \( 1 - 10.5iT - 841T^{2} \)
31 \( 1 + 52.3iT - 961T^{2} \)
37 \( 1 + 10.5T + 1.36e3T^{2} \)
41 \( 1 - 42.6iT - 1.68e3T^{2} \)
43 \( 1 + 45.9iT - 1.84e3T^{2} \)
47 \( 1 - 4.22iT - 2.20e3T^{2} \)
53 \( 1 - 95.8T + 2.80e3T^{2} \)
59 \( 1 - 6.38iT - 3.48e3T^{2} \)
61 \( 1 - 14.8T + 3.72e3T^{2} \)
67 \( 1 + 38.0T + 4.48e3T^{2} \)
71 \( 1 - 125. iT - 5.04e3T^{2} \)
73 \( 1 + 18.5iT - 5.32e3T^{2} \)
79 \( 1 - 137. iT - 6.24e3T^{2} \)
83 \( 1 + 127. iT - 6.88e3T^{2} \)
89 \( 1 - 19.8iT - 7.92e3T^{2} \)
97 \( 1 + 42.4T + 9.40e3T^{2} \)
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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

−10.86271188126755618560311959225, −9.871795843976143507962810239005, −8.665369702581405469896527647910, −7.18402175840859115448817975846, −6.57492923958254171338750872270, −5.60780184942583192898484401186, −4.58237173536124806173149472656, −3.97848967954764164924523472584, −2.57956744409694690937677482415, −0.42807977687899424949468772081, 2.15873416658989767308004772662, 3.35043652930821746736369949053, 4.59411398929968378208664280577, 5.47221897210253896663766749545, 6.04897416750698336364234743896, 6.89392349646175150898408263725, 8.522396354311146218459755308949, 9.238897111687614729688423973363, 10.59069237382813505378515223057, 11.58901091448357308215034236231

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