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Properties

Label 2-1475-1475.1179-c0-0-0
Degree $2$
Conductor $1475$
Sign $-0.929 - 0.368i$
Analytic cond. $0.736120$
Root an. cond. $0.857974$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.690 + 0.951i)3-s + (0.809 + 0.587i)4-s + (−0.809 + 0.587i)5-s + 1.17i·7-s + (−0.118 − 0.363i)9-s + (−1.11 + 0.363i)12-s − 1.17i·15-s + (0.309 + 0.951i)16-s + (0.5 − 0.363i)19-s − 20-s + (−1.11 − 0.812i)21-s + (0.309 − 0.951i)25-s + (−0.690 − 0.224i)27-s + (−0.690 + 0.951i)28-s + (0.5 + 0.363i)29-s + ⋯
L(s)  = 1  + (−0.690 + 0.951i)3-s + (0.809 + 0.587i)4-s + (−0.809 + 0.587i)5-s + 1.17i·7-s + (−0.118 − 0.363i)9-s + (−1.11 + 0.363i)12-s − 1.17i·15-s + (0.309 + 0.951i)16-s + (0.5 − 0.363i)19-s − 20-s + (−1.11 − 0.812i)21-s + (0.309 − 0.951i)25-s + (−0.690 − 0.224i)27-s + (−0.690 + 0.951i)28-s + (0.5 + 0.363i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1475\)    =    \(5^{2} \cdot 59\)
Sign: $-0.929 - 0.368i$
Analytic conductor: \(0.736120\)
Root analytic conductor: \(0.857974\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1475} (1179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1475,\ (\ :0),\ -0.929 - 0.368i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8557062906\)
\(L(\frac12)\) \(\approx\) \(0.8557062906\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (-0.309 - 0.951i)T \)
good2 \( 1 + (-0.809 - 0.587i)T^{2} \)
3 \( 1 + (0.690 - 0.951i)T + (-0.309 - 0.951i)T^{2} \)
7 \( 1 - 1.17iT - T^{2} \)
11 \( 1 + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
23 \( 1 + (-0.809 - 0.587i)T^{2} \)
29 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.618 + 1.90i)T + (-0.809 + 0.587i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.309 + 0.951i)T^{2} \)
53 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (0.809 + 0.587i)T^{2} \)
67 \( 1 + (0.309 - 0.951i)T^{2} \)
71 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (-0.809 - 0.587i)T^{2} \)
79 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (0.309 - 0.951i)T^{2} \)
89 \( 1 + (0.809 + 0.587i)T^{2} \)
97 \( 1 + (0.309 + 0.951i)T^{2} \)
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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

−10.39791571920305048062298215670, −9.234263792233858122673747942843, −8.455825336021422807480357774868, −7.58138722625155852327531990582, −6.81224290637327387519407087627, −5.90619583858772937001470462926, −5.10463132095373122857738080508, −4.05433172452703795835299969836, −3.19402645275567005261120625671, −2.26984174650609890307335561539, 0.78034784630732443848444784175, 1.59360252435402964415232911818, 3.23081316072780160007966810378, 4.36086837365120374554982008235, 5.32476604451072378158123356583, 6.27108346274457644481867393120, 6.93575561944087608235047381432, 7.54427289141650798876178893911, 8.175035367379058820998064319702, 9.511334656967699922754175736848

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