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Properties

Label 2-475-95.64-c1-0-6
Degree $2$
Conductor $475$
Sign $-0.819 - 0.573i$
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  + (−0.269 + 0.155i)2-s + (0.891 − 0.514i)3-s + (−0.951 + 1.64i)4-s + (−0.160 + 0.277i)6-s + 3.28i·7-s − 1.21i·8-s + (−0.969 + 1.67i)9-s − 5.16·11-s + 1.95i·12-s + (−3.06 − 1.76i)13-s + (−0.510 − 0.883i)14-s + (−1.71 − 2.96i)16-s + (0.874 − 0.504i)17-s − 0.603i·18-s + (−2.42 − 3.62i)19-s + ⋯
L(s)  = 1  + (−0.190 + 0.109i)2-s + (0.514 − 0.297i)3-s + (−0.475 + 0.824i)4-s + (−0.0653 + 0.113i)6-s + 1.23i·7-s − 0.429i·8-s + (−0.323 + 0.559i)9-s − 1.55·11-s + 0.565i·12-s + (−0.849 − 0.490i)13-s + (−0.136 − 0.236i)14-s + (−0.428 − 0.742i)16-s + (0.211 − 0.122i)17-s − 0.142i·18-s + (−0.555 − 0.831i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.819 - 0.573i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.819 - 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.819 - 0.573i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ -0.819 - 0.573i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.220628 + 0.700119i\)
\(L(\frac12)\) \(\approx\) \(0.220628 + 0.700119i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.42 + 3.62i)T \)
good2 \( 1 + (0.269 - 0.155i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-0.891 + 0.514i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 - 3.28iT - 7T^{2} \)
11 \( 1 + 5.16T + 11T^{2} \)
13 \( 1 + (3.06 + 1.76i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.874 + 0.504i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-6.63 - 3.83i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.01 - 3.48i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4.60T + 31T^{2} \)
37 \( 1 - 6.48iT - 37T^{2} \)
41 \( 1 + (-3.40 - 5.89i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.46 + 3.15i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.32 + 1.92i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.16 - 3.55i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.73 - 11.6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.06 - 5.31i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.69 + 5.59i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.227 - 0.394i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.57 + 2.06i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.44 - 2.50i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.50iT - 83T^{2} \)
89 \( 1 + (3.56 - 6.17i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.37 + 5.41i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.43461418071828606407184042948, −10.41833188546261852909693960686, −9.181389962096717256624514734801, −8.700853895018264311499742325656, −7.76879680486052634742750653487, −7.25627223701437834491164729502, −5.48221835646698947508762507936, −4.90020290698321711412066175444, −3.00064685939319892623512951345, −2.53075229983253141101896516717, 0.42590710299137820856125102397, 2.33825318603157539083246447956, 3.82000759642114418339440059886, 4.78284986556490320107301995868, 5.82753036850954361618300698482, 7.10474463878237531918861215959, 8.035931631928376319811298674654, 9.025655642342769665232655075392, 9.828120077417364870100112286756, 10.50472974178019628897099493178

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