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Properties

Label 2-475-19.5-c1-0-9
Degree $2$
Conductor $475$
Sign $0.0179 - 0.999i$
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  + (−2.17 + 0.792i)2-s + (−0.363 + 2.05i)3-s + (2.57 − 2.16i)4-s + (−0.840 − 4.76i)6-s + (−1.49 − 2.58i)7-s + (−1.58 + 2.74i)8-s + (−1.28 − 0.468i)9-s + (1.03 − 1.78i)11-s + (3.51 + 6.09i)12-s + (0.487 + 2.76i)13-s + (5.30 + 4.45i)14-s + (0.103 − 0.588i)16-s + (7.66 − 2.78i)17-s + 3.17·18-s + (−2.39 + 3.64i)19-s + ⋯
L(s)  = 1  + (−1.53 + 0.560i)2-s + (−0.209 + 1.18i)3-s + (1.28 − 1.08i)4-s + (−0.343 − 1.94i)6-s + (−0.565 − 0.978i)7-s + (−0.559 + 0.968i)8-s + (−0.429 − 0.156i)9-s + (0.311 − 0.539i)11-s + (1.01 + 1.75i)12-s + (0.135 + 0.766i)13-s + (1.41 + 1.18i)14-s + (0.0259 − 0.147i)16-s + (1.85 − 0.676i)17-s + 0.748·18-s + (−0.548 + 0.836i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0179 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0179 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.0179 - 0.999i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (176, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.0179 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.435736 + 0.427997i\)
\(L(\frac12)\) \(\approx\) \(0.435736 + 0.427997i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.39 - 3.64i)T \)
good2 \( 1 + (2.17 - 0.792i)T + (1.53 - 1.28i)T^{2} \)
3 \( 1 + (0.363 - 2.05i)T + (-2.81 - 1.02i)T^{2} \)
7 \( 1 + (1.49 + 2.58i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.03 + 1.78i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.487 - 2.76i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (-7.66 + 2.78i)T + (13.0 - 10.9i)T^{2} \)
23 \( 1 + (-3.38 + 2.84i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-7.67 - 2.79i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (2.50 + 4.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 1.47T + 37T^{2} \)
41 \( 1 + (1.17 - 6.67i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-5.25 - 4.41i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (-0.751 - 0.273i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (7.46 - 6.26i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (-8.51 + 3.09i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (1.57 - 1.31i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-1.52 - 0.553i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-4.14 - 3.47i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (2.54 - 14.4i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (-2.16 + 12.2i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (3.85 + 6.67i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.0972 + 0.551i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (-14.0 + 5.13i)T + (74.3 - 62.3i)T^{2} \)
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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

−10.68577596808422518242082714868, −10.08434964129676001294775533148, −9.602007341554644726229481680857, −8.730423097089875361409016111423, −7.72379142414044414510486557631, −6.84077612339173988128496921302, −5.89805663513300930358132012518, −4.48088616978045957307239766086, −3.39605052001348354490321011989, −1.03437843098512939597824000644, 0.873638494925587004230044025825, 2.06146144730590012474673298087, 3.18636726170007981948023608706, 5.44244844968307670191356630878, 6.53906591230938558280280491012, 7.37599685292295928846387533240, 8.174238817785521761123027277587, 8.976264875205469798746162418353, 9.863226193950314165691705967539, 10.60581975234949262364030525434

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