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Properties

Label 2-475-19.4-c1-0-23
Degree $2$
Conductor $475$
Sign $-0.349 - 0.936i$
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  + (−2.22 − 0.810i)2-s + (−0.396 − 2.25i)3-s + (2.77 + 2.32i)4-s + (−0.940 + 5.33i)6-s + (0.818 − 1.41i)7-s + (−1.92 − 3.32i)8-s + (−2.08 + 0.759i)9-s + (−1.36 − 2.36i)11-s + (4.13 − 7.16i)12-s + (−1.07 + 6.10i)13-s + (−2.97 + 2.49i)14-s + (0.325 + 1.84i)16-s + (−2.93 − 1.06i)17-s + 5.26·18-s + (−3.46 − 2.64i)19-s + ⋯
L(s)  = 1  + (−1.57 − 0.573i)2-s + (−0.229 − 1.29i)3-s + (1.38 + 1.16i)4-s + (−0.384 + 2.17i)6-s + (0.309 − 0.535i)7-s + (−0.679 − 1.17i)8-s + (−0.695 + 0.253i)9-s + (−0.411 − 0.712i)11-s + (1.19 − 2.06i)12-s + (−0.298 + 1.69i)13-s + (−0.794 + 0.666i)14-s + (0.0812 + 0.460i)16-s + (−0.712 − 0.259i)17-s + 1.24·18-s + (−0.795 − 0.605i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.349 - 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.349 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.349 - 0.936i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ -0.349 - 0.936i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.109164 + 0.157317i\)
\(L(\frac12)\) \(\approx\) \(0.109164 + 0.157317i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.46 + 2.64i)T \)
good2 \( 1 + (2.22 + 0.810i)T + (1.53 + 1.28i)T^{2} \)
3 \( 1 + (0.396 + 2.25i)T + (-2.81 + 1.02i)T^{2} \)
7 \( 1 + (-0.818 + 1.41i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.36 + 2.36i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.07 - 6.10i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (2.93 + 1.06i)T + (13.0 + 10.9i)T^{2} \)
23 \( 1 + (5.59 + 4.69i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (2.09 - 0.761i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (2.21 - 3.83i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.04T + 37T^{2} \)
41 \( 1 + (-0.681 - 3.86i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (0.362 - 0.303i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (-2.16 + 0.787i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (-4.87 - 4.09i)T + (9.20 + 52.1i)T^{2} \)
59 \( 1 + (11.5 + 4.19i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (3.66 + 3.07i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (0.630 - 0.229i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (-2.01 + 1.69i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (-1.13 - 6.41i)T + (-68.5 + 24.9i)T^{2} \)
79 \( 1 + (-0.715 - 4.05i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-3.21 + 5.56i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3.00 + 17.0i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (0.122 + 0.0444i)T + (74.3 + 62.3i)T^{2} \)
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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

−10.53005485354505313560255681224, −9.343182430324746606936318038553, −8.601822034162134484255221154783, −7.76597776639572141376160983691, −6.99508168824975886787817657292, −6.31834528941157466394726752109, −4.41160827398997762560339334135, −2.47520777230246050831393687883, −1.60766224118373378493990377877, −0.19304745275256490517375736220, 2.13769328892338819856316469187, 3.96382058094886550612455051173, 5.28969286495020741711229518036, 6.03050783125012469817775648705, 7.50694566293183933056147778152, 8.111037271736413925497629755810, 9.083362248282897327764550847319, 9.822930221227191108296071834169, 10.43015610697354753261354027221, 10.94796093543863336224295154753

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