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Properties

Label 2-475-95.94-c2-0-26
Degree $2$
Conductor $475$
Sign $0.679 - 0.734i$
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  + 3.38·2-s − 4.08·3-s + 7.43·4-s − 13.8·6-s + 4.56i·7-s + 11.6·8-s + 7.71·9-s + 6.24·11-s − 30.4·12-s + 14.4·13-s + 15.4i·14-s + 9.57·16-s + 26.6i·17-s + 26.0·18-s + (18.2 + 5.30i)19-s + ⋯
L(s)  = 1  + 1.69·2-s − 1.36·3-s + 1.85·4-s − 2.30·6-s + 0.652i·7-s + 1.45·8-s + 0.857·9-s + 0.567·11-s − 2.53·12-s + 1.11·13-s + 1.10i·14-s + 0.598·16-s + 1.57i·17-s + 1.44·18-s + (0.960 + 0.279i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.679 - 0.734i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.679 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.679 - 0.734i$
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: \((2,\ 475,\ (\ :1),\ 0.679 - 0.734i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.055035909\)
\(L(\frac12)\) \(\approx\) \(3.055035909\)
\(L(2)\) 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 \( 1 + (-18.2 - 5.30i)T \)
good2 \( 1 - 3.38T + 4T^{2} \)
3 \( 1 + 4.08T + 9T^{2} \)
7 \( 1 - 4.56iT - 49T^{2} \)
11 \( 1 - 6.24T + 121T^{2} \)
13 \( 1 - 14.4T + 169T^{2} \)
17 \( 1 - 26.6iT - 289T^{2} \)
23 \( 1 - 33.4iT - 529T^{2} \)
29 \( 1 + 10.3iT - 841T^{2} \)
31 \( 1 + 30.5iT - 961T^{2} \)
37 \( 1 - 18.1T + 1.36e3T^{2} \)
41 \( 1 - 65.5iT - 1.68e3T^{2} \)
43 \( 1 + 71.3iT - 1.84e3T^{2} \)
47 \( 1 - 53.1iT - 2.20e3T^{2} \)
53 \( 1 + 21.9T + 2.80e3T^{2} \)
59 \( 1 + 92.9iT - 3.48e3T^{2} \)
61 \( 1 + 27.8T + 3.72e3T^{2} \)
67 \( 1 + 60.7T + 4.48e3T^{2} \)
71 \( 1 + 103. iT - 5.04e3T^{2} \)
73 \( 1 - 11.9iT - 5.32e3T^{2} \)
79 \( 1 + 9.41iT - 6.24e3T^{2} \)
83 \( 1 - 15.2iT - 6.88e3T^{2} \)
89 \( 1 + 11.8iT - 7.92e3T^{2} \)
97 \( 1 - 62.0T + 9.40e3T^{2} \)
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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

−11.43030631172609518393955514775, −10.59767777071565953472133999112, −9.281825401766096801420046126067, −7.84697224052061015256917891390, −6.46260342818331095950908752749, −5.97314812162427806697652918970, −5.42944070039758996630109650574, −4.27045875163257770265192217240, −3.35796378482730986201911064440, −1.57636592559153464784971520741, 0.943173104919839290203118079147, 2.97389572138439932521646692177, 4.17121751231227190487245575976, 4.95791856691106789888504252988, 5.77041922065000203040794393816, 6.64195800516089610437586769771, 7.20921963171059613660633160954, 8.949474480468543069868310898000, 10.39160735142511745325889856910, 11.08602436178364411399791947042

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