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Properties

Label 2-2300-115.114-c2-0-23
Degree $2$
Conductor $2300$
Sign $-0.447 - 0.894i$
Analytic cond. $62.6704$
Root an. cond. $7.91646$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.89i·3-s − 6.27·7-s − 6.14·9-s + 11.1i·11-s − 15.9i·13-s + 23.9·17-s − 7.55i·19-s − 24.4i·21-s + (18.3 + 13.8i)23-s + 11.1i·27-s − 10.3·29-s + 53.0·31-s − 43.4·33-s + 26.7·37-s + 62.1·39-s + ⋯
L(s)  = 1  + 1.29i·3-s − 0.896·7-s − 0.682·9-s + 1.01i·11-s − 1.22i·13-s + 1.40·17-s − 0.397i·19-s − 1.16i·21-s + (0.799 + 0.600i)23-s + 0.411i·27-s − 0.356·29-s + 1.71·31-s − 1.31·33-s + 0.723·37-s + 1.59·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2300\)    =    \(2^{2} \cdot 5^{2} \cdot 23\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(62.6704\)
Root analytic conductor: \(7.91646\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2300} (1149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2300,\ (\ :1),\ -0.447 - 0.894i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.841402285\)
\(L(\frac12)\) \(\approx\) \(1.841402285\)
\(L(2)\) 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 \( 1 + (-18.3 - 13.8i)T \)
good3 \( 1 - 3.89iT - 9T^{2} \)
7 \( 1 + 6.27T + 49T^{2} \)
11 \( 1 - 11.1iT - 121T^{2} \)
13 \( 1 + 15.9iT - 169T^{2} \)
17 \( 1 - 23.9T + 289T^{2} \)
19 \( 1 + 7.55iT - 361T^{2} \)
29 \( 1 + 10.3T + 841T^{2} \)
31 \( 1 - 53.0T + 961T^{2} \)
37 \( 1 - 26.7T + 1.36e3T^{2} \)
41 \( 1 - 39.5T + 1.68e3T^{2} \)
43 \( 1 + 11.0T + 1.84e3T^{2} \)
47 \( 1 + 56.1iT - 2.20e3T^{2} \)
53 \( 1 + 92.4T + 2.80e3T^{2} \)
59 \( 1 - 78.5T + 3.48e3T^{2} \)
61 \( 1 - 113. iT - 3.72e3T^{2} \)
67 \( 1 - 55.8T + 4.48e3T^{2} \)
71 \( 1 - 59.4T + 5.04e3T^{2} \)
73 \( 1 - 9.67iT - 5.32e3T^{2} \)
79 \( 1 - 75.4iT - 6.24e3T^{2} \)
83 \( 1 + 141.T + 6.88e3T^{2} \)
89 \( 1 - 21.7iT - 7.92e3T^{2} \)
97 \( 1 + 19.6T + 9.40e3T^{2} \)
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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

−9.459430992584372306441383919715, −8.402277426432528535536845602378, −7.57502090490104148233945964826, −6.76015430321437879037647872925, −5.68047156741508532622234144214, −5.10544334901848239400043996883, −4.22132192901538453970895913733, −3.36256338304882585533315605623, −2.72746936828795101485850014484, −0.963704908666223074659992426534, 0.57701314637621019387050764699, 1.42758381296817360561034504140, 2.64660730427554347295811330556, 3.40801367481886330685214483471, 4.55362270639950776781086137987, 5.80376063626882882605653284170, 6.39076589984133212016080168333, 6.87134347001986833192298881150, 7.86041016664777254361948962305, 8.305126847225587349823610223294

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