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Properties

Label 2-2300-460.367-c0-0-11
Degree $2$
Conductor $2300$
Sign $0.874 + 0.485i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 + 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 + 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2300\)    =    \(2^{2} \cdot 5^{2} \cdot 23\)
Sign: $0.874 + 0.485i$
Analytic conductor: \(1.14784\)
Root analytic conductor: \(1.07137\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2300} (2207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2300,\ (\ :0),\ 0.874 + 0.485i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7314760872\)
\(L(\frac12)\) \(\approx\) \(0.7314760872\)
\(L(1)\) 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

−8.982946856124163444891962426364, −8.300341941238969786745895028152, −7.45702294200861540626771670251, −6.78726605334438017788214858593, −6.11674386415106972314156320919, −5.45605388265775995258976277074, −4.34683090199868934636645577277, −3.76388349950111102000042969354, −2.22463493411257141622054360349, −0.56434950799444421953839733563, 1.41757506899496246008676726682, 2.51118336993937423126920242022, 3.39883447741491177843938119068, 4.41654753645013546431061821356, 5.14232243922328123258508458023, 5.77163854241281162501758484940, 7.15903357374033714241500871828, 7.86681228806563962433651943737, 8.654641436860103970473905463095, 9.390770959748007902445242088097

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