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Properties

Label 2-475-95.54-c1-0-11
Degree $2$
Conductor $475$
Sign $0.866 - 0.499i$
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  + (0.443 − 0.528i)2-s + (0.237 + 0.652i)3-s + (0.264 + 1.50i)4-s + (0.450 + 0.164i)6-s + (2.02 − 1.16i)7-s + (2.10 + 1.21i)8-s + (1.92 − 1.61i)9-s + (−2.28 + 3.96i)11-s + (−0.916 + 0.529i)12-s + (0.438 − 1.20i)13-s + (0.279 − 1.58i)14-s + (−1.28 + 0.468i)16-s + (−0.420 + 0.501i)17-s − 1.73i·18-s + (−3.67 + 2.34i)19-s + ⋯
L(s)  = 1  + (0.313 − 0.373i)2-s + (0.137 + 0.376i)3-s + (0.132 + 0.750i)4-s + (0.183 + 0.0669i)6-s + (0.764 − 0.441i)7-s + (0.744 + 0.429i)8-s + (0.642 − 0.539i)9-s + (−0.690 + 1.19i)11-s + (−0.264 + 0.152i)12-s + (0.121 − 0.333i)13-s + (0.0747 − 0.424i)14-s + (−0.321 + 0.117i)16-s + (−0.102 + 0.121i)17-s − 0.409i·18-s + (−0.843 + 0.537i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.499i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 - 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.866 - 0.499i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.866 - 0.499i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.91678 + 0.512857i\)
\(L(\frac12)\) \(\approx\) \(1.91678 + 0.512857i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.67 - 2.34i)T \)
good2 \( 1 + (-0.443 + 0.528i)T + (-0.347 - 1.96i)T^{2} \)
3 \( 1 + (-0.237 - 0.652i)T + (-2.29 + 1.92i)T^{2} \)
7 \( 1 + (-2.02 + 1.16i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.28 - 3.96i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.438 + 1.20i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (0.420 - 0.501i)T + (-2.95 - 16.7i)T^{2} \)
23 \( 1 + (-5.48 + 0.966i)T + (21.6 - 7.86i)T^{2} \)
29 \( 1 + (-3.62 + 3.04i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-2.24 - 3.88i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 7.79iT - 37T^{2} \)
41 \( 1 + (8.17 - 2.97i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-9.44 - 1.66i)T + (40.4 + 14.7i)T^{2} \)
47 \( 1 + (4.06 + 4.84i)T + (-8.16 + 46.2i)T^{2} \)
53 \( 1 + (6.50 - 1.14i)T + (49.8 - 18.1i)T^{2} \)
59 \( 1 + (4.51 + 3.78i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (1.30 + 7.38i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (8.39 + 10.0i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (-0.651 + 3.69i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (2.72 + 7.48i)T + (-55.9 + 46.9i)T^{2} \)
79 \( 1 + (-5.92 + 2.15i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (8.51 - 4.91i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-11.4 - 4.16i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-2.70 + 3.22i)T + (-16.8 - 95.5i)T^{2} \)
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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

−10.89298140209174897879284602104, −10.49519913781694223243012388481, −9.381486003454109799331642639314, −8.245760751060062082676294199761, −7.55113151790109775327331985802, −6.62812793314852879923560356835, −4.84399468387937007542347612559, −4.36510897010272128358562660453, −3.18876054968438641624137721541, −1.82186785947037871621243893294, 1.31573504457652731982273229075, 2.64566438049546723610054973046, 4.52329081439057653677706662022, 5.22498755971517153792954992639, 6.26550136869883261181383311680, 7.15286983622855411504696532904, 8.160903510311048185661735461598, 8.955491782236445199293471810572, 10.26599133085405131046471085268, 10.90452427737267794548089736900

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