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Properties

Label 2-475-475.111-c1-0-42
Degree $2$
Conductor $475$
Sign $-0.724 + 0.689i$
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  + (1.18 + 0.0828i)2-s + (−0.0989 − 2.83i)3-s + (−0.583 − 0.0819i)4-s + (0.980 + 2.00i)5-s + (0.117 − 3.36i)6-s + (−0.798 − 1.38i)7-s + (−3.00 − 0.639i)8-s + (−5.03 + 0.351i)9-s + (0.995 + 2.46i)10-s + (−0.530 − 5.04i)11-s + (−0.174 + 1.66i)12-s + (−1.08 − 1.60i)13-s + (−0.831 − 1.70i)14-s + (5.59 − 2.97i)15-s + (−2.37 − 0.682i)16-s + (−0.130 + 0.322i)17-s + ⋯
L(s)  = 1  + (0.837 + 0.0585i)2-s + (−0.0571 − 1.63i)3-s + (−0.291 − 0.0409i)4-s + (0.438 + 0.898i)5-s + (0.0480 − 1.37i)6-s + (−0.301 − 0.522i)7-s + (−1.06 − 0.226i)8-s + (−1.67 + 0.117i)9-s + (0.314 + 0.778i)10-s + (−0.159 − 1.52i)11-s + (−0.0504 + 0.479i)12-s + (−0.300 − 0.445i)13-s + (−0.222 − 0.455i)14-s + (1.44 − 0.769i)15-s + (−0.594 − 0.170i)16-s + (−0.0316 + 0.0782i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.724 + 0.689i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.724 + 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.724 + 0.689i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ -0.724 + 0.689i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.563922 - 1.41076i\)
\(L(\frac12)\) \(\approx\) \(0.563922 - 1.41076i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.980 - 2.00i)T \)
19 \( 1 + (-2.94 + 3.21i)T \)
good2 \( 1 + (-1.18 - 0.0828i)T + (1.98 + 0.278i)T^{2} \)
3 \( 1 + (0.0989 + 2.83i)T + (-2.99 + 0.209i)T^{2} \)
7 \( 1 + (0.798 + 1.38i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.530 + 5.04i)T + (-10.7 + 2.28i)T^{2} \)
13 \( 1 + (1.08 + 1.60i)T + (-4.86 + 12.0i)T^{2} \)
17 \( 1 + (0.130 - 0.322i)T + (-12.2 - 11.8i)T^{2} \)
23 \( 1 + (0.439 + 0.424i)T + (0.802 + 22.9i)T^{2} \)
29 \( 1 + (-2.26 - 5.61i)T + (-20.8 + 20.1i)T^{2} \)
31 \( 1 + (-1.03 + 1.15i)T + (-3.24 - 30.8i)T^{2} \)
37 \( 1 + (-4.45 - 3.23i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-8.36 - 2.39i)T + (34.7 + 21.7i)T^{2} \)
43 \( 1 + (-0.00348 + 0.0197i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (2.82 + 6.99i)T + (-33.8 + 32.6i)T^{2} \)
53 \( 1 + (7.50 + 1.05i)T + (50.9 + 14.6i)T^{2} \)
59 \( 1 + (1.83 + 7.37i)T + (-52.0 + 27.6i)T^{2} \)
61 \( 1 + (-8.21 - 7.93i)T + (2.12 + 60.9i)T^{2} \)
67 \( 1 + (3.43 + 2.14i)T + (29.3 + 60.2i)T^{2} \)
71 \( 1 + (-12.5 - 6.66i)T + (39.7 + 58.8i)T^{2} \)
73 \( 1 + (-7.15 + 10.6i)T + (-27.3 - 67.6i)T^{2} \)
79 \( 1 + (0.123 + 3.54i)T + (-78.8 + 5.51i)T^{2} \)
83 \( 1 + (4.06 - 4.51i)T + (-8.67 - 82.5i)T^{2} \)
89 \( 1 + (7.82 - 2.24i)T + (75.4 - 47.1i)T^{2} \)
97 \( 1 + (-12.9 + 8.07i)T + (42.5 - 87.1i)T^{2} \)
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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

−11.01283432258794352548993351625, −9.829453852839187966921447196215, −8.648869431216023937594369061245, −7.67462038521979774844515365729, −6.70184427684644652605441933728, −6.13738183188515055528884633301, −5.24893334268128201654880173991, −3.43672665241199546705278898863, −2.67733448604266610906506382018, −0.72324854811257650386389814580, 2.54823159397766899460091856389, 3.99482610472312816750089104553, 4.58392421177558144332244001428, 5.28382119813360425766032252066, 6.10971771175880039970398874832, 7.958761957204408618251120614227, 9.243954589127949801369186122145, 9.449402131570916679759663934127, 10.13614553659639975044099899890, 11.51688205233777335175722606548

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