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Properties

Label 2-2300-460.183-c0-0-5
Degree $2$
Conductor $2300$
Sign $-0.300 - 0.953i$
Analytic cond. $1.14784$
Root an. cond. $1.07137$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.0871 + 0.996i)2-s + (0.245 − 0.245i)3-s + (−0.984 + 0.173i)4-s + (0.266 + 0.223i)6-s + (−0.258 − 0.965i)8-s + 0.879i·9-s + (−0.199 + 0.284i)12-s + (0.483 + 0.483i)13-s + (0.939 − 0.342i)16-s + (−0.876 + 0.0766i)18-s + (0.707 − 0.707i)23-s + (−0.300 − 0.173i)24-s + (−0.439 + 0.524i)26-s + (0.461 + 0.461i)27-s + 1.53i·29-s + ⋯
L(s)  = 1  + (0.0871 + 0.996i)2-s + (0.245 − 0.245i)3-s + (−0.984 + 0.173i)4-s + (0.266 + 0.223i)6-s + (−0.258 − 0.965i)8-s + 0.879i·9-s + (−0.199 + 0.284i)12-s + (0.483 + 0.483i)13-s + (0.939 − 0.342i)16-s + (−0.876 + 0.0766i)18-s + (0.707 − 0.707i)23-s + (−0.300 − 0.173i)24-s + (−0.439 + 0.524i)26-s + (0.461 + 0.461i)27-s + 1.53i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.300 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.300 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2300\)    =    \(2^{2} \cdot 5^{2} \cdot 23\)
Sign: $-0.300 - 0.953i$
Analytic conductor: \(1.14784\)
Root analytic conductor: \(1.07137\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2300} (643, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2300,\ (\ :0),\ -0.300 - 0.953i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.181771388\)
\(L(\frac12)\) \(\approx\) \(1.181771388\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0871 - 0.996i)T \)
5 \( 1 \)
23 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 + (-0.245 + 0.245i)T - iT^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (-0.483 - 0.483i)T + iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 - T^{2} \)
29 \( 1 - 1.53iT - T^{2} \)
31 \( 1 - 1.28iT - T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + 0.347T + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (1.32 + 1.32i)T + iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 - 1.73T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 - 0.684iT - T^{2} \)
73 \( 1 + (-0.909 - 0.909i)T + iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + iT^{2} \)
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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

−8.988853253119716451338181021464, −8.594288757578689947463090401443, −7.894830077496204340376970549108, −6.91471067639575834382507388586, −6.66690542628363896970398756472, −5.37037287168359267577607519335, −4.94437507784385905304339739301, −3.89114857218672741722665773690, −2.89388640672522159336280829187, −1.45926989591128924251528870152, 0.859336781691396772559674989693, 2.20463298975037962128310989053, 3.25012301682448155235491259210, 3.83413623762635534060283755700, 4.73888362881599139750990604003, 5.71356596281385633394207763540, 6.44624214524486711640709854303, 7.71466131427883415861330291064, 8.391738339562037883282352058258, 9.213167275217866261841118870453

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