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Properties

Label 2-475-95.64-c1-0-13
Degree $2$
Conductor $475$
Sign $0.641 - 0.767i$
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  + (0.950 − 0.548i)2-s + (0.328 − 0.189i)3-s + (−0.397 + 0.689i)4-s + (0.208 − 0.360i)6-s + 1.89i·7-s + 3.06i·8-s + (−1.42 + 2.47i)9-s + 0.134·11-s + 0.301i·12-s + (3.04 + 1.75i)13-s + (1.03 + 1.79i)14-s + (0.887 + 1.53i)16-s + (−1.43 + 0.830i)17-s + 3.13i·18-s + (−2.10 − 3.81i)19-s + ⋯
L(s)  = 1  + (0.672 − 0.388i)2-s + (0.189 − 0.109i)3-s + (−0.198 + 0.344i)4-s + (0.0849 − 0.147i)6-s + 0.715i·7-s + 1.08i·8-s + (−0.476 + 0.824i)9-s + 0.0405·11-s + 0.0871i·12-s + (0.843 + 0.487i)13-s + (0.277 + 0.480i)14-s + (0.221 + 0.384i)16-s + (−0.348 + 0.201i)17-s + 0.738i·18-s + (−0.483 − 0.875i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.641 - 0.767i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.641 - 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.641 - 0.767i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.641 - 0.767i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.66841 + 0.780302i\)
\(L(\frac12)\) \(\approx\) \(1.66841 + 0.780302i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 + (2.10 + 3.81i)T \)
good2 \( 1 + (-0.950 + 0.548i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-0.328 + 0.189i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 - 1.89iT - 7T^{2} \)
11 \( 1 - 0.134T + 11T^{2} \)
13 \( 1 + (-3.04 - 1.75i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.43 - 0.830i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-4.65 - 2.68i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.48 + 4.30i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.56T + 31T^{2} \)
37 \( 1 + 1.69iT - 37T^{2} \)
41 \( 1 + (5.31 + 9.20i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (7.36 - 4.25i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-9.62 - 5.55i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.229 + 0.132i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.44 + 5.97i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.58 - 7.94i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.55 - 1.47i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.664 + 1.15i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.49 + 3.17i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.733 + 1.27i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 7.44iT - 83T^{2} \)
89 \( 1 + (-4.86 + 8.43i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-15.1 + 8.73i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.37259430126930388864555013184, −10.55675427312163725961308126178, −8.968970211305603800070720163258, −8.656442998493318326474877939172, −7.61751127808572066601245525272, −6.30154810092315487270352529006, −5.23433738816325047608459755785, −4.35717122339405510658071405974, −3.07845209317665553219980119779, −2.14958811639758940628369202044, 0.961766580424582837667672865557, 3.21677857207363501647950811476, 4.10508382166596380410414596263, 5.14920808202233975113753020402, 6.27806233404564315386034463575, 6.81077956087569311739697384189, 8.237055025870552864850413967260, 9.036296312807081771200290632612, 10.11648909184908974563618394687, 10.70608768533886866738309526131

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