Properties

Label 2-1575-21.20-c1-0-13
Degree $2$
Conductor $1575$
Sign $-0.537 - 0.843i$
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  + 0.517i·2-s + 1.73·4-s + (−1 + 2.44i)7-s + 1.93i·8-s + 0.378i·11-s + 1.79i·13-s + (−1.26 − 0.517i)14-s + 2.46·16-s − 3.46·17-s + 1.79i·19-s − 0.196·22-s + 1.41i·23-s − 0.928·26-s + (−1.73 + 4.24i)28-s − 1.41i·29-s + ⋯
L(s)  = 1  + 0.366i·2-s + 0.866·4-s + (−0.377 + 0.925i)7-s + 0.683i·8-s + 0.114i·11-s + 0.497i·13-s + (−0.338 − 0.138i)14-s + 0.616·16-s − 0.840·17-s + 0.411i·19-s − 0.0418·22-s + 0.294i·23-s − 0.182·26-s + (−0.327 + 0.801i)28-s − 0.262i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.537 - 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.537 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.537 - 0.843i$
Analytic conductor: \(12.5764\)
Root analytic conductor: \(3.54632\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1575,\ (\ :1/2),\ -0.537 - 0.843i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.687922679\)
\(L(\frac12)\) \(\approx\) \(1.687922679\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 2.44i)T \)
good2 \( 1 - 0.517iT - 2T^{2} \)
11 \( 1 - 0.378iT - 11T^{2} \)
13 \( 1 - 1.79iT - 13T^{2} \)
17 \( 1 + 3.46T + 17T^{2} \)
19 \( 1 - 1.79iT - 19T^{2} \)
23 \( 1 - 1.41iT - 23T^{2} \)
29 \( 1 + 1.41iT - 29T^{2} \)
31 \( 1 - 6.69iT - 31T^{2} \)
37 \( 1 - 1.46T + 37T^{2} \)
41 \( 1 + 10.3T + 41T^{2} \)
43 \( 1 - 4.92T + 43T^{2} \)
47 \( 1 + 9.46T + 47T^{2} \)
53 \( 1 + 10.1iT - 53T^{2} \)
59 \( 1 - 9.46T + 59T^{2} \)
61 \( 1 - 13.3iT - 61T^{2} \)
67 \( 1 - 10.9T + 67T^{2} \)
71 \( 1 - 15.0iT - 71T^{2} \)
73 \( 1 + 1.79iT - 73T^{2} \)
79 \( 1 + 10.9T + 79T^{2} \)
83 \( 1 - 9.46T + 83T^{2} \)
89 \( 1 - 12.9T + 89T^{2} \)
97 \( 1 - 15.1iT - 97T^{2} \)
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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

−9.694871595924946930839742592456, −8.700918763503966056084567345465, −8.208125394775006009215791099101, −7.01459636118908847318828430890, −6.61819148397085737027265794916, −5.72641893913960476278511251009, −4.97237481236319717081153515247, −3.63244661424104735421847718263, −2.59766341470026924219244363866, −1.73884013747167011942458183318, 0.60692318059298194346119915080, 1.97788079613858379426440910948, 3.03936652351935202388385817938, 3.85619590074714188386106387303, 4.90145846275400143737148333785, 6.14516664035955040737070642987, 6.71448721601969620544894681179, 7.47873852813941432752181179660, 8.242271791760575878122383453188, 9.381527166829061580367596824802

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