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Properties

Label 2-475-95.49-c1-0-19
Degree $2$
Conductor $475$
Sign $-0.943 - 0.330i$
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  + (−1.44 − 0.832i)2-s + (1.00 + 0.579i)3-s + (0.385 + 0.667i)4-s + (−0.965 − 1.67i)6-s + 2.43i·7-s + 2.04i·8-s + (−0.827 − 1.43i)9-s − 5.75·11-s + 0.893i·12-s + (−1.38 + 0.797i)13-s + (2.02 − 3.51i)14-s + (2.47 − 4.28i)16-s + (−5.18 − 2.99i)17-s + 2.75i·18-s + (−0.149 − 4.35i)19-s + ⋯
L(s)  = 1  + (−1.01 − 0.588i)2-s + (0.579 + 0.334i)3-s + (0.192 + 0.333i)4-s + (−0.394 − 0.682i)6-s + 0.920i·7-s + 0.723i·8-s + (−0.275 − 0.477i)9-s − 1.73·11-s + 0.258i·12-s + (−0.383 + 0.221i)13-s + (0.541 − 0.938i)14-s + (0.618 − 1.07i)16-s + (−1.25 − 0.725i)17-s + 0.649i·18-s + (−0.0342 − 0.999i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 - 0.330i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.943 - 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.943 - 0.330i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ -0.943 - 0.330i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00510256 + 0.0300485i\)
\(L(\frac12)\) \(\approx\) \(0.00510256 + 0.0300485i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.149 + 4.35i)T \)
good2 \( 1 + (1.44 + 0.832i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-1.00 - 0.579i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 - 2.43iT - 7T^{2} \)
11 \( 1 + 5.75T + 11T^{2} \)
13 \( 1 + (1.38 - 0.797i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (5.18 + 2.99i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (0.814 - 0.470i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.30 - 2.26i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.26T + 31T^{2} \)
37 \( 1 - 2.89iT - 37T^{2} \)
41 \( 1 + (-3.15 + 5.46i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.93 + 2.26i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (7.75 - 4.47i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.90 - 1.09i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.39 - 9.35i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.26 - 9.11i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.874 - 0.504i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.41 - 7.65i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-8.87 - 5.12i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.80 + 6.58i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.11iT - 83T^{2} \)
89 \( 1 + (5.55 + 9.62i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.51 - 2.02i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.38013952486403109493348769705, −9.444463655621156966248131361038, −8.924629919775339260582156324077, −8.276435178791435807383732751907, −7.15654032364244764319934405816, −5.64702515873563194030179715603, −4.75237727435314868531806683921, −2.86252484843042626590007242381, −2.32975020296729982922550012618, −0.02205667776929365591382112343, 2.06365640218063522582199525100, 3.53437728329305282810130407295, 4.91874118590898479416109916758, 6.30541148095741616490584501036, 7.40230483558580269121967482986, 7.956972127929004685036227449820, 8.425966685420537352817363541885, 9.629517950266527181771976444888, 10.43936126255174876820807925641, 11.01166331615071229093634741924

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