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Properties

Label 2-1575-15.2-c1-0-1
Degree $2$
Conductor $1575$
Sign $-0.662 + 0.749i$
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  + (−0.876 + 0.876i)2-s + 0.464i·4-s + (0.707 + 0.707i)7-s + (−2.15 − 2.15i)8-s + 4.66i·11-s + (−1.54 + 1.54i)13-s − 1.23·14-s + 2.85·16-s + (−2.32 + 2.32i)17-s − 3.54i·19-s + (−4.08 − 4.08i)22-s + (−1.44 − 1.44i)23-s − 2.71i·26-s + (−0.328 + 0.328i)28-s − 7.58·29-s + ⋯
L(s)  = 1  + (−0.619 + 0.619i)2-s + 0.232i·4-s + (0.267 + 0.267i)7-s + (−0.763 − 0.763i)8-s + 1.40i·11-s + (−0.429 + 0.429i)13-s − 0.331·14-s + 0.713·16-s + (−0.565 + 0.565i)17-s − 0.813i·19-s + (−0.870 − 0.870i)22-s + (−0.300 − 0.300i)23-s − 0.532i·26-s + (−0.0620 + 0.0620i)28-s − 1.40·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.662 + 0.749i$
Analytic conductor: \(12.5764\)
Root analytic conductor: \(3.54632\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (1457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1575,\ (\ :1/2),\ -0.662 + 0.749i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3812347544\)
\(L(\frac12)\) \(\approx\) \(0.3812347544\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.876 - 0.876i)T - 2iT^{2} \)
11 \( 1 - 4.66iT - 11T^{2} \)
13 \( 1 + (1.54 - 1.54i)T - 13iT^{2} \)
17 \( 1 + (2.32 - 2.32i)T - 17iT^{2} \)
19 \( 1 + 3.54iT - 19T^{2} \)
23 \( 1 + (1.44 + 1.44i)T + 23iT^{2} \)
29 \( 1 + 7.58T + 29T^{2} \)
31 \( 1 - 8.12T + 31T^{2} \)
37 \( 1 + (-6.00 - 6.00i)T + 37iT^{2} \)
41 \( 1 + 6.72iT - 41T^{2} \)
43 \( 1 + (5.46 - 5.46i)T - 43iT^{2} \)
47 \( 1 + (5.16 - 5.16i)T - 47iT^{2} \)
53 \( 1 + (6.86 + 6.86i)T + 53iT^{2} \)
59 \( 1 + 6.40T + 59T^{2} \)
61 \( 1 + 13.6T + 61T^{2} \)
67 \( 1 + (2.77 + 2.77i)T + 67iT^{2} \)
71 \( 1 + 0.576iT - 71T^{2} \)
73 \( 1 + (-4.94 + 4.94i)T - 73iT^{2} \)
79 \( 1 + 2.50iT - 79T^{2} \)
83 \( 1 + (-3.99 - 3.99i)T + 83iT^{2} \)
89 \( 1 - 6.86T + 89T^{2} \)
97 \( 1 + (4.43 + 4.43i)T + 97iT^{2} \)
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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

−9.492928346882827347070824757352, −9.266672768912213040453300185122, −8.143290588529029985408277675814, −7.70148374368234270897657256093, −6.75792362204359369012281662306, −6.29051108957546606847105709006, −4.87843848941323596227930893258, −4.25991496047938067566120938546, −2.92428389531327404699729078501, −1.81850118354582125250360404091, 0.18330694315185295069848401144, 1.38094630255968198479815281574, 2.57873209285399111718773648538, 3.51091216042756513750894112495, 4.77973900105204184424322466714, 5.71913709977593745964078997735, 6.32296691449186749907478716615, 7.63646052633534735765819189974, 8.244848468647259240088187863515, 9.073867633143155519113861352921

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