Properties

Label 2-2300-115.22-c1-0-21
Degree $2$
Conductor $2300$
Sign $0.938 + 0.345i$
Analytic cond. $18.3655$
Root an. cond. $4.28550$
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  + (1.87 − 1.87i)3-s + (−1.41 + 1.41i)7-s − 4i·9-s + 5.29i·11-s + (1.87 − 1.87i)13-s + 5.29·19-s + 5.29i·21-s + (0.957 + 4.69i)23-s + (−1.87 − 1.87i)27-s + i·29-s + 3·31-s + (9.89 + 9.89i)33-s + (7.07 − 7.07i)37-s − 7i·39-s + 9·41-s + ⋯
L(s)  = 1  + (1.08 − 1.08i)3-s + (−0.534 + 0.534i)7-s − 1.33i·9-s + 1.59i·11-s + (0.518 − 0.518i)13-s + 1.21·19-s + 1.15i·21-s + (0.199 + 0.979i)23-s + (−0.360 − 0.360i)27-s + 0.185i·29-s + 0.538·31-s + (1.72 + 1.72i)33-s + (1.16 − 1.16i)37-s − 1.12i·39-s + 1.40·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 + 0.345i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.938 + 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2300\)    =    \(2^{2} \cdot 5^{2} \cdot 23\)
Sign: $0.938 + 0.345i$
Analytic conductor: \(18.3655\)
Root analytic conductor: \(4.28550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2300} (1057, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2300,\ (\ :1/2),\ 0.938 + 0.345i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.637068906\)
\(L(\frac12)\) \(\approx\) \(2.637068906\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 + (-0.957 - 4.69i)T \)
good3 \( 1 + (-1.87 + 1.87i)T - 3iT^{2} \)
7 \( 1 + (1.41 - 1.41i)T - 7iT^{2} \)
11 \( 1 - 5.29iT - 11T^{2} \)
13 \( 1 + (-1.87 + 1.87i)T - 13iT^{2} \)
17 \( 1 - 17iT^{2} \)
19 \( 1 - 5.29T + 19T^{2} \)
29 \( 1 - iT - 29T^{2} \)
31 \( 1 - 3T + 31T^{2} \)
37 \( 1 + (-7.07 + 7.07i)T - 37iT^{2} \)
41 \( 1 - 9T + 41T^{2} \)
43 \( 1 + (4.24 + 4.24i)T + 43iT^{2} \)
47 \( 1 + (1.87 + 1.87i)T + 47iT^{2} \)
53 \( 1 + (2.82 + 2.82i)T + 53iT^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 - 5.29iT - 61T^{2} \)
67 \( 1 + (-1.41 + 1.41i)T - 67iT^{2} \)
71 \( 1 + 9T + 71T^{2} \)
73 \( 1 + (-9.35 + 9.35i)T - 73iT^{2} \)
79 \( 1 - 5.29T + 79T^{2} \)
83 \( 1 + (-8.48 - 8.48i)T + 83iT^{2} \)
89 \( 1 + 15.8T + 89T^{2} \)
97 \( 1 + (-1.41 + 1.41i)T - 97iT^{2} \)
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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

−9.087658124755222198052899740507, −7.935040906005482710636703044245, −7.57652395271108286913932840597, −6.87967359962724506365155514943, −6.00338794199987178252513132316, −5.06254965884393502083720484057, −3.81435575515360248779456649049, −2.92779798875699446705692387366, −2.18449062216632006279645032143, −1.16895900925532800959835261838, 0.951093365315849331947592801084, 2.74236224879773729351677789384, 3.26843363156486795727331908212, 4.02115758975093772207136491423, 4.80590231714070933935978262587, 5.95973792544123700893523592153, 6.65687055575892683207122646676, 7.87015153378100251521540490538, 8.369001847408232100997695930410, 9.125297386306260065533407026241

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