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Properties

Label 2-2300-460.107-c0-0-1
Degree $2$
Conductor $2300$
Sign $0.482 + 0.875i$
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  + (−0.599 − 0.800i)2-s + (−0.0994 + 0.266i)3-s + (−0.281 + 0.959i)4-s + (0.273 − 0.0801i)6-s + (−0.229 − 1.05i)7-s + (0.936 − 0.349i)8-s + (0.694 + 0.601i)9-s + (−0.227 − 0.170i)12-s + (−0.708 + 0.817i)14-s + (−0.841 − 0.540i)16-s + (0.0655 − 0.916i)18-s + (0.304 + 0.0437i)21-s + (0.800 − 0.599i)23-s + 0.284i·24-s + (−0.479 + 0.261i)27-s + (1.07 + 0.0771i)28-s + ⋯
L(s)  = 1  + (−0.599 − 0.800i)2-s + (−0.0994 + 0.266i)3-s + (−0.281 + 0.959i)4-s + (0.273 − 0.0801i)6-s + (−0.229 − 1.05i)7-s + (0.936 − 0.349i)8-s + (0.694 + 0.601i)9-s + (−0.227 − 0.170i)12-s + (−0.708 + 0.817i)14-s + (−0.841 − 0.540i)16-s + (0.0655 − 0.916i)18-s + (0.304 + 0.0437i)21-s + (0.800 − 0.599i)23-s + 0.284i·24-s + (−0.479 + 0.261i)27-s + (1.07 + 0.0771i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.482 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.482 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2300\)    =    \(2^{2} \cdot 5^{2} \cdot 23\)
Sign: $0.482 + 0.875i$
Analytic conductor: \(1.14784\)
Root analytic conductor: \(1.07137\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2300} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2300,\ (\ :0),\ 0.482 + 0.875i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8465286333\)
\(L(\frac12)\) \(\approx\) \(0.8465286333\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.599 + 0.800i)T \)
5 \( 1 \)
23 \( 1 + (-0.800 + 0.599i)T \)
good3 \( 1 + (0.0994 - 0.266i)T + (-0.755 - 0.654i)T^{2} \)
7 \( 1 + (0.229 + 1.05i)T + (-0.909 + 0.415i)T^{2} \)
11 \( 1 + (-0.959 + 0.281i)T^{2} \)
13 \( 1 + (-0.909 - 0.415i)T^{2} \)
17 \( 1 + (-0.540 - 0.841i)T^{2} \)
19 \( 1 + (-0.841 - 0.540i)T^{2} \)
29 \( 1 + (0.368 + 1.25i)T + (-0.841 + 0.540i)T^{2} \)
31 \( 1 + (0.654 + 0.755i)T^{2} \)
37 \( 1 + (-0.989 + 0.142i)T^{2} \)
41 \( 1 + (-1.10 - 1.27i)T + (-0.142 + 0.989i)T^{2} \)
43 \( 1 + (-0.527 - 0.196i)T + (0.755 + 0.654i)T^{2} \)
47 \( 1 + (-0.926 + 0.926i)T - iT^{2} \)
53 \( 1 + (-0.909 + 0.415i)T^{2} \)
59 \( 1 + (0.415 - 0.909i)T^{2} \)
61 \( 1 + (-0.512 - 0.234i)T + (0.654 + 0.755i)T^{2} \)
67 \( 1 + (1.45 - 1.09i)T + (0.281 - 0.959i)T^{2} \)
71 \( 1 + (0.959 + 0.281i)T^{2} \)
73 \( 1 + (0.540 - 0.841i)T^{2} \)
79 \( 1 + (-0.415 + 0.909i)T^{2} \)
83 \( 1 + (0.129 + 1.81i)T + (-0.989 + 0.142i)T^{2} \)
89 \( 1 + (0.822 + 1.80i)T + (-0.654 + 0.755i)T^{2} \)
97 \( 1 + (0.989 + 0.142i)T^{2} \)
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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

−9.226526398278506651123912135449, −8.422236897600992240154425899682, −7.42731226631707750848809867882, −7.22880321533782092669020208170, −5.97054388616970857346425787777, −4.58553574035851384852627970573, −4.25607100111509664217110719304, −3.22130545783352666138881874200, −2.13925059384780492207143316297, −0.899325009357386814416457721055, 1.16149011974783649326319270338, 2.34621185054888464303446382015, 3.68951320409067132709046173351, 4.83245229690179204238683669982, 5.66787445618878679061439010760, 6.23970218475727896642694500456, 7.15557133194075516432999483613, 7.57508199186462752041344047174, 8.775887143034856428517190702318, 9.097035126189410492756408075181

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