Properties

Label 2-1575-21.20-c1-0-43
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  − 1.93i·2-s − 1.73·4-s + (−1 + 2.44i)7-s − 0.517i·8-s + 5.27i·11-s − 6.69i·13-s + (4.73 + 1.93i)14-s − 4.46·16-s − 3.46·17-s − 6.69i·19-s + 10.1·22-s + 1.41i·23-s − 12.9·26-s + (1.73 − 4.24i)28-s − 1.41i·29-s + ⋯
L(s)  = 1  − 1.36i·2-s − 0.866·4-s + (−0.377 + 0.925i)7-s − 0.183i·8-s + 1.59i·11-s − 1.85i·13-s + (1.26 + 0.516i)14-s − 1.11·16-s − 0.840·17-s − 1.53i·19-s + 2.17·22-s + 0.294i·23-s − 2.53·26-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

\(L(1)\) \(\approx\) \(0.4734771314\)
\(L(\frac12)\) \(\approx\) \(0.4734771314\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.93iT - 2T^{2} \)
11 \( 1 - 5.27iT - 11T^{2} \)
13 \( 1 + 6.69iT - 13T^{2} \)
17 \( 1 + 3.46T + 17T^{2} \)
19 \( 1 + 6.69iT - 19T^{2} \)
23 \( 1 - 1.41iT - 23T^{2} \)
29 \( 1 + 1.41iT - 29T^{2} \)
31 \( 1 + 1.79iT - 31T^{2} \)
37 \( 1 + 5.46T + 37T^{2} \)
41 \( 1 + 10.3T + 41T^{2} \)
43 \( 1 + 8.92T + 43T^{2} \)
47 \( 1 - 2.53T + 47T^{2} \)
53 \( 1 - 4.52iT - 53T^{2} \)
59 \( 1 + 2.53T + 59T^{2} \)
61 \( 1 + 3.58iT - 61T^{2} \)
67 \( 1 + 2.92T + 67T^{2} \)
71 \( 1 + 9.41iT - 71T^{2} \)
73 \( 1 - 6.69iT - 73T^{2} \)
79 \( 1 - 2.92T + 79T^{2} \)
83 \( 1 + 2.53T + 83T^{2} \)
89 \( 1 - 0.928T + 89T^{2} \)
97 \( 1 + 10.2iT - 97T^{2} \)
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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

−9.201927693424201935724673481350, −8.410965395416398106653598114942, −7.25892733723252795743822814651, −6.53513597041654909270423705602, −5.27288602014921514800326649622, −4.61617643212068070660401879610, −3.35805011163688987684933260444, −2.64007372771614772705347750599, −1.82296152483570410160246725543, −0.17114046287340713083768345047, 1.73492033069647161129645230219, 3.40572981782655680787615359705, 4.23574759849997614264635815688, 5.22155650966124867975715477809, 6.25842290666714857585715401016, 6.63579756358362180508736033291, 7.33932506714161068177982705374, 8.449522191807926573076185829626, 8.677009781068355670986560715271, 9.748138296944242623515507445543

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