Properties

Label 2-475-475.11-c1-0-4
Degree $2$
Conductor $475$
Sign $0.0467 + 0.998i$
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  + (0.267 + 2.54i)2-s + (1.17 + 1.30i)3-s + (−4.43 + 0.942i)4-s + (−0.956 − 2.02i)5-s + (−3.01 + 3.34i)6-s − 3.81·7-s + (−2.00 − 6.15i)8-s + (−0.0102 + 0.0976i)9-s + (4.88 − 2.97i)10-s + (−3.34 + 2.43i)11-s + (−6.45 − 4.69i)12-s + (−0.373 + 3.55i)13-s + (−1.01 − 9.69i)14-s + (1.51 − 3.63i)15-s + (6.83 − 3.04i)16-s + (3.40 + 0.723i)17-s + ⋯
L(s)  = 1  + (0.188 + 1.79i)2-s + (0.679 + 0.755i)3-s + (−2.21 + 0.471i)4-s + (−0.427 − 0.903i)5-s + (−1.22 + 1.36i)6-s − 1.44·7-s + (−0.707 − 2.17i)8-s + (−0.00342 + 0.0325i)9-s + (1.54 − 0.939i)10-s + (−1.00 + 0.732i)11-s + (−1.86 − 1.35i)12-s + (−0.103 + 0.986i)13-s + (−0.272 − 2.59i)14-s + (0.391 − 0.937i)15-s + (1.70 − 0.760i)16-s + (0.825 + 0.175i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0467 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0467 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.0467 + 0.998i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.0467 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.438256 - 0.418228i\)
\(L(\frac12)\) \(\approx\) \(0.438256 - 0.418228i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.956 + 2.02i)T \)
19 \( 1 + (4.24 + 1.00i)T \)
good2 \( 1 + (-0.267 - 2.54i)T + (-1.95 + 0.415i)T^{2} \)
3 \( 1 + (-1.17 - 1.30i)T + (-0.313 + 2.98i)T^{2} \)
7 \( 1 + 3.81T + 7T^{2} \)
11 \( 1 + (3.34 - 2.43i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (0.373 - 3.55i)T + (-12.7 - 2.70i)T^{2} \)
17 \( 1 + (-3.40 - 0.723i)T + (15.5 + 6.91i)T^{2} \)
23 \( 1 + (-1.14 - 0.508i)T + (15.3 + 17.0i)T^{2} \)
29 \( 1 + (-2.10 + 0.447i)T + (26.4 - 11.7i)T^{2} \)
31 \( 1 + (-1.43 - 4.41i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-0.365 - 0.265i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (10.7 - 4.80i)T + (27.4 - 30.4i)T^{2} \)
43 \( 1 + (4.30 - 7.45i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.703 - 0.149i)T + (42.9 - 19.1i)T^{2} \)
53 \( 1 + (-1.30 + 0.278i)T + (48.4 - 21.5i)T^{2} \)
59 \( 1 + (11.2 - 5.01i)T + (39.4 - 43.8i)T^{2} \)
61 \( 1 + (-11.2 - 5.00i)T + (40.8 + 45.3i)T^{2} \)
67 \( 1 + (-1.15 + 1.28i)T + (-7.00 - 66.6i)T^{2} \)
71 \( 1 + (-9.25 - 10.2i)T + (-7.42 + 70.6i)T^{2} \)
73 \( 1 + (1.37 + 13.0i)T + (-71.4 + 15.1i)T^{2} \)
79 \( 1 + (8.52 + 9.46i)T + (-8.25 + 78.5i)T^{2} \)
83 \( 1 + (-0.631 - 1.94i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-4.75 - 2.11i)T + (59.5 + 66.1i)T^{2} \)
97 \( 1 + (1.81 + 2.01i)T + (-10.1 + 96.4i)T^{2} \)
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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

−12.11185144161896814052135994355, −10.11665315190551997409258745398, −9.562319602699078267617351720870, −8.789191218396440753036762417558, −8.134626344717670935842018642898, −7.04438469491358894155368545132, −6.27293336505021327483100500032, −4.99988669251428813641882174509, −4.32867585868809937058818008710, −3.28066054745660899639558445007, 0.30813892082065585804798429380, 2.33349468047232689872849690842, 3.05378527687104474580006820381, 3.61043123387820708382765414987, 5.35355919768515763534090718752, 6.67747720180668829362851140076, 7.889221491858584638082005388020, 8.615411872638470949991579312476, 9.991725424226166072733419825429, 10.32800666116085875407551947845

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