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Properties

Label 2-2300-115.114-c2-0-5
Degree $2$
Conductor $2300$
Sign $-0.689 + 0.724i$
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  + 3.10i·3-s + 1.98·7-s − 0.662·9-s + 10.6i·11-s − 1.75i·13-s − 21.2·17-s − 4.57i·19-s + 6.16i·21-s + (6.72 − 21.9i)23-s + 25.9i·27-s − 9.21·29-s − 42.5·31-s − 33.2·33-s + 63.8·37-s + 5.45·39-s + ⋯
L(s)  = 1  + 1.03i·3-s + 0.283·7-s − 0.0736·9-s + 0.971i·11-s − 0.135i·13-s − 1.25·17-s − 0.240i·19-s + 0.293i·21-s + (0.292 − 0.956i)23-s + 0.959i·27-s − 0.317·29-s − 1.37·31-s − 1.00·33-s + 1.72·37-s + 0.139·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.689 + 0.724i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.689 + 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2300\)    =    \(2^{2} \cdot 5^{2} \cdot 23\)
Sign: $-0.689 + 0.724i$
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: \((2,\ 2300,\ (\ :1),\ -0.689 + 0.724i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.3731980926\)
\(L(\frac12)\) \(\approx\) \(0.3731980926\)
\(L(2)\) 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 \( 1 + (-6.72 + 21.9i)T \)
good3 \( 1 - 3.10iT - 9T^{2} \)
7 \( 1 - 1.98T + 49T^{2} \)
11 \( 1 - 10.6iT - 121T^{2} \)
13 \( 1 + 1.75iT - 169T^{2} \)
17 \( 1 + 21.2T + 289T^{2} \)
19 \( 1 + 4.57iT - 361T^{2} \)
29 \( 1 + 9.21T + 841T^{2} \)
31 \( 1 + 42.5T + 961T^{2} \)
37 \( 1 - 63.8T + 1.36e3T^{2} \)
41 \( 1 + 22.0T + 1.68e3T^{2} \)
43 \( 1 + 61.7T + 1.84e3T^{2} \)
47 \( 1 - 20.7iT - 2.20e3T^{2} \)
53 \( 1 - 55.7T + 2.80e3T^{2} \)
59 \( 1 + 34.0T + 3.48e3T^{2} \)
61 \( 1 - 76.1iT - 3.72e3T^{2} \)
67 \( 1 + 64.4T + 4.48e3T^{2} \)
71 \( 1 + 126.T + 5.04e3T^{2} \)
73 \( 1 + 103. iT - 5.32e3T^{2} \)
79 \( 1 - 39.5iT - 6.24e3T^{2} \)
83 \( 1 - 117.T + 6.88e3T^{2} \)
89 \( 1 + 78.6iT - 7.92e3T^{2} \)
97 \( 1 - 74.1T + 9.40e3T^{2} \)
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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

−9.234079005870728200619609830204, −8.862168396193256021347171867937, −7.75240489117976605942146953996, −7.03189171048768785553059630780, −6.18410290040206711978250614004, −5.01203615083251867246956301250, −4.58115096148541026589203654794, −3.85360866390952761130544506524, −2.69244249579572661317154613233, −1.63666074356134140786693776458, 0.086884467670964162326948945681, 1.32767315936757000531918328001, 2.12172126699960571760622851398, 3.27658460213180990751032030554, 4.26507916959573488506566731698, 5.31260647269633322973664969596, 6.15479164578833379452049055133, 6.82467374026439096535077559548, 7.55848462818645965772626304771, 8.226877074977235892077098450448

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