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Properties

Label 2-475-95.88-c1-0-20
Degree $2$
Conductor $475$
Sign $0.760 + 0.649i$
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  + (0.366 − 1.36i)2-s + (2.73 + 0.732i)3-s + (2 − 3.46i)6-s + (2 − 1.99i)8-s + (4.33 + 2.5i)9-s − 3·11-s + (1.09 + 4.09i)13-s + (−1.99 − 3.46i)16-s + (−2.36 − 0.633i)17-s + (5 − 4.99i)18-s + (−4.33 − 0.5i)19-s + (−1.09 + 4.09i)22-s + (−7.09 + 1.90i)23-s + (6.92 − 3.99i)24-s + 6·26-s + (3.99 + 4i)27-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)2-s + (1.57 + 0.422i)3-s + (0.816 − 1.41i)6-s + (0.707 − 0.707i)8-s + (1.44 + 0.833i)9-s − 0.904·11-s + (0.304 + 1.13i)13-s + (−0.499 − 0.866i)16-s + (−0.573 − 0.153i)17-s + (1.17 − 1.17i)18-s + (−0.993 − 0.114i)19-s + (−0.234 + 0.873i)22-s + (−1.48 + 0.396i)23-s + (1.41 − 0.816i)24-s + 1.17·26-s + (0.769 + 0.769i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.760 + 0.649i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.760 + 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.760 + 0.649i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (468, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.760 + 0.649i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.61667 - 0.964773i\)
\(L(\frac12)\) \(\approx\) \(2.61667 - 0.964773i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.33 + 0.5i)T \)
good2 \( 1 + (-0.366 + 1.36i)T + (-1.73 - i)T^{2} \)
3 \( 1 + (-2.73 - 0.732i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 - 7iT^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + (-1.09 - 4.09i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (2.36 + 0.633i)T + (14.7 + 8.5i)T^{2} \)
23 \( 1 + (7.09 - 1.90i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-2.59 + 4.5i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 8.66iT - 31T^{2} \)
37 \( 1 + (-3 - 3i)T + 37iT^{2} \)
41 \( 1 + (-9 + 5.19i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.90 - 7.09i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (0.633 + 2.36i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.732 + 2.73i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.59 + 4.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-12.2 + 3.29i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (13.5 - 7.79i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.90 - 7.09i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (2.59 + 4.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.66 - 8.66i)T + 83iT^{2} \)
89 \( 1 + (-2.59 + 4.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.29 - 12.2i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.89971266520939385407439915752, −9.923530401989053724697081600082, −9.382362579837750380785392552940, −8.245625111747697334246489924310, −7.63297603532514218889793185738, −6.34070666427693890469144506102, −4.38863847387964240320789321788, −3.94716443586955813978666307376, −2.60591435391699141092600566572, −2.07372184048013591112592595224, 1.96364042412231031425943918880, 2.98394444640065145132130579166, 4.36689803182823206975354108339, 5.65835495029115522496897562276, 6.67306122610439644492351843064, 7.61492993608371721631430428168, 8.223257467519063919992805204980, 8.775964699772257689629129071995, 10.21700352732196173806817078995, 10.79838373550552429838001192073

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