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Properties

Label 2-1575-15.8-c1-0-12
Degree $2$
Conductor $1575$
Sign $0.374 + 0.927i$
Analytic cond. $12.5764$
Root an. cond. $3.54632$
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  + (−1.36 − 1.36i)2-s + 1.73i·4-s + (0.707 − 0.707i)7-s + (−0.366 + 0.366i)8-s + 2.44i·11-s + (−1.03 − 1.03i)13-s − 1.93·14-s + 4.46·16-s + (−2 − 2i)17-s + 2i·19-s + (3.34 − 3.34i)22-s + (−0.464 + 0.464i)23-s + 2.82i·26-s + (1.22 + 1.22i)28-s − 3.48·29-s + ⋯
L(s)  = 1  + (−0.965 − 0.965i)2-s + 0.866i·4-s + (0.267 − 0.267i)7-s + (−0.129 + 0.129i)8-s + 0.738i·11-s + (−0.287 − 0.287i)13-s − 0.516·14-s + 1.11·16-s + (−0.485 − 0.485i)17-s + 0.458i·19-s + (0.713 − 0.713i)22-s + (−0.0967 + 0.0967i)23-s + 0.554i·26-s + (0.231 + 0.231i)28-s − 0.647·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.374 + 0.927i$
Analytic conductor: \(12.5764\)
Root analytic conductor: \(3.54632\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (1268, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1575,\ (\ :1/2),\ 0.374 + 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9102377181\)
\(L(\frac12)\) \(\approx\) \(0.9102377181\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 + (-0.707 + 0.707i)T \)
good2 \( 1 + (1.36 + 1.36i)T + 2iT^{2} \)
11 \( 1 - 2.44iT - 11T^{2} \)
13 \( 1 + (1.03 + 1.03i)T + 13iT^{2} \)
17 \( 1 + (2 + 2i)T + 17iT^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
23 \( 1 + (0.464 - 0.464i)T - 23iT^{2} \)
29 \( 1 + 3.48T + 29T^{2} \)
31 \( 1 - 8.92T + 31T^{2} \)
37 \( 1 + (-5.27 + 5.27i)T - 37iT^{2} \)
41 \( 1 - 7.72iT - 41T^{2} \)
43 \( 1 + (-2.82 - 2.82i)T + 43iT^{2} \)
47 \( 1 + (0.535 + 0.535i)T + 47iT^{2} \)
53 \( 1 + (-5.73 + 5.73i)T - 53iT^{2} \)
59 \( 1 - 6.96T + 59T^{2} \)
61 \( 1 - 4.92T + 61T^{2} \)
67 \( 1 + (3.48 - 3.48i)T - 67iT^{2} \)
71 \( 1 - 10.1iT - 71T^{2} \)
73 \( 1 + (-3.86 - 3.86i)T + 73iT^{2} \)
79 \( 1 + 4iT - 79T^{2} \)
83 \( 1 + (-11.4 + 11.4i)T - 83iT^{2} \)
89 \( 1 - 14.1T + 89T^{2} \)
97 \( 1 + (13.6 - 13.6i)T - 97iT^{2} \)
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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

−9.596662549317742136908720041378, −8.607299417463563109924598838362, −7.929368438303295238749230698135, −7.16083723797253899191510486625, −6.05656816759607372208867537393, −5.00996064011703454153362655445, −4.03547245010320536408250581531, −2.81118344612023631643728108482, −1.98876225556966800831592543864, −0.77556165323936913298098558578, 0.77512274773794949832601444266, 2.36632382563614786072983961433, 3.63462562363320829961359492673, 4.79527161625179501018786946290, 5.85632949235188371264183042281, 6.45477816136810360626677634837, 7.28895928843672926131908609334, 8.081985131297771053392242642658, 8.675773143630006107005831176645, 9.272647228858552158732571302135

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