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Properties

Label 2-1575-105.89-c1-0-16
Degree $2$
Conductor $1575$
Sign $-0.644 - 0.764i$
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  + (−1.23 + 2.13i)2-s + (−2.04 − 3.53i)4-s + (1.88 + 1.85i)7-s + 5.14·8-s + (−2.50 + 1.44i)11-s + 5.72·13-s + (−6.28 + 1.74i)14-s + (−2.25 + 3.91i)16-s + (−2.54 + 1.46i)17-s + (3.13 + 1.81i)19-s − 7.12i·22-s + (3.96 − 6.87i)23-s + (−7.06 + 12.2i)26-s + (2.69 − 10.4i)28-s − 5.38i·29-s + ⋯
L(s)  = 1  + (−0.872 + 1.51i)2-s + (−1.02 − 1.76i)4-s + (0.713 + 0.700i)7-s + 1.81·8-s + (−0.754 + 0.435i)11-s + 1.58·13-s + (−1.68 + 0.467i)14-s + (−0.564 + 0.978i)16-s + (−0.616 + 0.356i)17-s + (0.719 + 0.415i)19-s − 1.51i·22-s + (0.827 − 1.43i)23-s + (−1.38 + 2.40i)26-s + (0.509 − 1.97i)28-s − 1.00i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.644 - 0.764i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.644 - 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.644 - 0.764i$
Analytic conductor: \(12.5764\)
Root analytic conductor: \(3.54632\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (1349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1575,\ (\ :1/2),\ -0.644 - 0.764i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.111318477\)
\(L(\frac12)\) \(\approx\) \(1.111318477\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.88 - 1.85i)T \)
good2 \( 1 + (1.23 - 2.13i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (2.50 - 1.44i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.72T + 13T^{2} \)
17 \( 1 + (2.54 - 1.46i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.13 - 1.81i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.96 + 6.87i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.38iT - 29T^{2} \)
31 \( 1 + (-0.435 + 0.251i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.82 - 1.63i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.54T + 41T^{2} \)
43 \( 1 + 10.1iT - 43T^{2} \)
47 \( 1 + (-6.55 - 3.78i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.03 - 8.72i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.28 - 5.69i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (9.67 + 5.58i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.39 + 1.38i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.14iT - 71T^{2} \)
73 \( 1 + (0.284 + 0.492i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.58 - 7.93i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 15.0iT - 83T^{2} \)
89 \( 1 + (-1.76 + 3.05i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 17.6T + 97T^{2} \)
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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

−9.213765660181414716160987122917, −8.776258231552475325188733107524, −8.092765722259806649849996460062, −7.51716556894770892432361904365, −6.48046219465254326760371297688, −5.88754797852690687062372280864, −5.13759758067142549980753318937, −4.18740755634870120691821038234, −2.47616826655530441287699107168, −1.02670731056210044515236417221, 0.798851438946441602934037778639, 1.62778356690398277997641283148, 2.94582902479892155462330868431, 3.63142396776227346156886948397, 4.65008115407491783201945257849, 5.71368972812526269865095216990, 7.11724327670893407139560246564, 7.85674989739556425589516064428, 8.634852470718415079609586910379, 9.173221484404584581926975873952

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