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Properties

Label 2-1575-105.59-c1-0-44
Degree $2$
Conductor $1575$
Sign $-0.421 + 0.906i$
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  + (0.707 + 1.22i)2-s + (−2.59 − 0.5i)7-s + 2.82·8-s + (−1.22 − 0.707i)11-s − 5.19·13-s + (−1.22 − 3.53i)14-s + (2.00 + 3.46i)16-s + (−4.24 − 2.44i)17-s + (−1.5 + 0.866i)19-s − 2i·22-s + (−2.82 − 4.89i)23-s + (−3.67 − 6.36i)26-s + 2.82i·29-s + (1.5 + 0.866i)31-s − 6.92i·34-s + ⋯
L(s)  = 1  + (0.499 + 0.866i)2-s + (−0.981 − 0.188i)7-s + 0.999·8-s + (−0.369 − 0.213i)11-s − 1.44·13-s + (−0.327 − 0.944i)14-s + (0.500 + 0.866i)16-s + (−1.02 − 0.594i)17-s + (−0.344 + 0.198i)19-s − 0.426i·22-s + (−0.589 − 1.02i)23-s + (−0.720 − 1.24i)26-s + 0.525i·29-s + (0.269 + 0.155i)31-s − 1.18i·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.421 + 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.421 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.421 + 0.906i$
Analytic conductor: \(12.5764\)
Root analytic conductor: \(3.54632\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (899, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1575,\ (\ :1/2),\ -0.421 + 0.906i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4236079601\)
\(L(\frac12)\) \(\approx\) \(0.4236079601\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.59 + 0.5i)T \)
good2 \( 1 + (-0.707 - 1.22i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (1.22 + 0.707i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.19T + 13T^{2} \)
17 \( 1 + (4.24 + 2.44i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.5 - 0.866i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.82 + 4.89i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.82iT - 29T^{2} \)
31 \( 1 + (-1.5 - 0.866i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.866 - 0.5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 7.34T + 41T^{2} \)
43 \( 1 - iT - 43T^{2} \)
47 \( 1 + (10.6 - 6.12i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.41 + 2.44i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.44 + 4.24i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3 - 1.73i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.52 + 5.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 7.07iT - 71T^{2} \)
73 \( 1 + (-0.866 + 1.5i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.5 - 4.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 7.34iT - 83T^{2} \)
89 \( 1 + (2.44 + 4.24i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 10.3T + 97T^{2} \)
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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

−9.153824123810545098295744157307, −8.149640075821297574144172934814, −7.29629585257422477952505959383, −6.66559208203759609692382184127, −6.10126203440332032064107578045, −4.98156958858118915349489467159, −4.49955143807767756689382102425, −3.16922010203841444705619912149, −2.08973064326958645177339882740, −0.12225798077449202764645261149, 1.95249731965899687470009127079, 2.66321864480005228017365175026, 3.61962244900445663104858370809, 4.47653500335332855007211543286, 5.35444771866857587973505752378, 6.50423614855448318282765644959, 7.22369833873794151393873181414, 8.045460413451546844376664440091, 9.075243907549676825274558517174, 10.04325211303463914869731155414

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