Properties

Label 2-7500-5.4-c1-0-24
Degree $2$
Conductor $7500$
Sign $-i$
Analytic cond. $59.8878$
Root an. cond. $7.73872$
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  + i·3-s − 0.747i·7-s − 9-s + 0.209·11-s − 2.50i·13-s + 6.81i·17-s − 1.24·19-s + 0.747·21-s + 3.25i·23-s i·27-s + 5.18·29-s + 3.73·31-s + 0.209i·33-s − 1.96i·37-s + 2.50·39-s + ⋯
L(s)  = 1  + 0.577i·3-s − 0.282i·7-s − 0.333·9-s + 0.0630·11-s − 0.694i·13-s + 1.65i·17-s − 0.285·19-s + 0.163·21-s + 0.677i·23-s − 0.192i·27-s + 0.962·29-s + 0.671·31-s + 0.0363i·33-s − 0.323i·37-s + 0.400·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7500\)    =    \(2^{2} \cdot 3 \cdot 5^{4}\)
Sign: $-i$
Analytic conductor: \(59.8878\)
Root analytic conductor: \(7.73872\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7500} (1249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 7500,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.650363540\)
\(L(\frac12)\) \(\approx\) \(1.650363540\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - iT \)
5 \( 1 \)
good7 \( 1 + 0.747iT - 7T^{2} \)
11 \( 1 - 0.209T + 11T^{2} \)
13 \( 1 + 2.50iT - 13T^{2} \)
17 \( 1 - 6.81iT - 17T^{2} \)
19 \( 1 + 1.24T + 19T^{2} \)
23 \( 1 - 3.25iT - 23T^{2} \)
29 \( 1 - 5.18T + 29T^{2} \)
31 \( 1 - 3.73T + 31T^{2} \)
37 \( 1 + 1.96iT - 37T^{2} \)
41 \( 1 - 3.21T + 41T^{2} \)
43 \( 1 + 12.7iT - 43T^{2} \)
47 \( 1 - 6.48iT - 47T^{2} \)
53 \( 1 - 4.14iT - 53T^{2} \)
59 \( 1 + 11.7T + 59T^{2} \)
61 \( 1 + 12.4T + 61T^{2} \)
67 \( 1 + 2.91iT - 67T^{2} \)
71 \( 1 - 6.55T + 71T^{2} \)
73 \( 1 + 2.51iT - 73T^{2} \)
79 \( 1 - 10.2T + 79T^{2} \)
83 \( 1 + 5.67iT - 83T^{2} \)
89 \( 1 - 0.921T + 89T^{2} \)
97 \( 1 - 15.5iT - 97T^{2} \)
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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

−7.998671026220578444218587192321, −7.58030890858699567402162748103, −6.48223936734841780920709762603, −6.00744439580916140424832084678, −5.23969616779960001421657174343, −4.41650383617104617423622218576, −3.79860519254192723015592413874, −3.07216068103367636387873485654, −2.06102239810051673830391937656, −0.943014542728720522841855419825, 0.46719554701026393159259236004, 1.52245246961098847665838456627, 2.56978697251351836306978245355, 3.02915785969883678441164815471, 4.33008232893092221770613993930, 4.80898281286861673458744653151, 5.70830122294491738999958327143, 6.53952869034718401389527761681, 6.85958178635551033340510103032, 7.74466924239375226317299687955

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