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Properties

Label 2-475-19.6-c1-0-9
Degree $2$
Conductor $475$
Sign $-0.836 - 0.548i$
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  + (−1.75 + 1.47i)2-s + (3.10 + 1.13i)3-s + (0.563 − 3.19i)4-s + (−7.11 + 2.59i)6-s + (−1.46 + 2.54i)7-s + (1.42 + 2.46i)8-s + (6.08 + 5.10i)9-s + (0.288 + 0.500i)11-s + (5.36 − 9.29i)12-s + (−0.629 + 0.229i)13-s + (−1.16 − 6.62i)14-s + (−0.0331 − 0.0120i)16-s + (0.269 − 0.226i)17-s − 18.1·18-s + (−3.86 + 2.02i)19-s + ⋯
L(s)  = 1  + (−1.24 + 1.04i)2-s + (1.79 + 0.653i)3-s + (0.281 − 1.59i)4-s + (−2.90 + 1.05i)6-s + (−0.555 + 0.962i)7-s + (0.503 + 0.872i)8-s + (2.02 + 1.70i)9-s + (0.0870 + 0.150i)11-s + (1.54 − 2.68i)12-s + (−0.174 + 0.0635i)13-s + (−0.312 − 1.77i)14-s + (−0.00829 − 0.00302i)16-s + (0.0653 − 0.0548i)17-s − 4.28·18-s + (−0.886 + 0.463i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.836 - 0.548i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.836 - 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.836 - 0.548i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ -0.836 - 0.548i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.362027 + 1.21239i\)
\(L(\frac12)\) \(\approx\) \(0.362027 + 1.21239i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.86 - 2.02i)T \)
good2 \( 1 + (1.75 - 1.47i)T + (0.347 - 1.96i)T^{2} \)
3 \( 1 + (-3.10 - 1.13i)T + (2.29 + 1.92i)T^{2} \)
7 \( 1 + (1.46 - 2.54i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.288 - 0.500i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.629 - 0.229i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (-0.269 + 0.226i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (-0.715 + 4.05i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (-2.30 - 1.93i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (0.148 - 0.257i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 8.30T + 37T^{2} \)
41 \( 1 + (2.51 + 0.913i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-1.14 - 6.49i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (8.45 + 7.09i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (-0.713 + 4.04i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (-0.467 + 0.392i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-0.178 + 1.01i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-2.55 - 2.14i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (2.29 + 13.0i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (-6.70 - 2.44i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (1.44 + 0.527i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-6.65 + 11.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-6.17 + 2.24i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (9.34 - 7.84i)T + (16.8 - 95.5i)T^{2} \)
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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

−10.71071371217155135648162633904, −9.819703652871706217096427662958, −9.416616106097822627696924385074, −8.522826180205496137817181267293, −8.227499887037281878716861891309, −7.14371714653422976054326866372, −6.16848392626724572216209009928, −4.68041437584372935148858685641, −3.27197987819747898155573474549, −2.08382692361860356792891476443, 0.990871839795019120223310474888, 2.24741314047570831038643726168, 3.17435934318569535650375786289, 4.05777958374734971089182261385, 6.61264135682410287755221670207, 7.50920153835499989212715291431, 8.138600671079904955427814330306, 8.974518176131964598832428601244, 9.649116455931070237164966626071, 10.26444091754776147891591095999

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