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Properties

Label 2-2300-460.43-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: $\chi_{2300} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2300,\ (\ :0),\ -0.482 + 0.875i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.625028647\)
\(L(\frac12)\) \(\approx\) \(1.625028647\)
\(L(1)\) 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

−9.199712412553743631390734989800, −8.322240411702360487604449909855, −7.18399420466483612443054071190, −6.63003137112996837179815816274, −5.57365492118436697870997706174, −4.66638758507873514910426852621, −3.97526676480559153643212638389, −3.40306851151679341907533823217, −2.07880945675344799234627047150, −0.952246930639936093608800518732, 1.91786975218527244430048509855, 2.81370828552173889471587434203, 4.00118039969511952311410615320, 4.77979985000586099858523211361, 5.60618387467426250444385468083, 6.23829095188732472930520309137, 7.11872335088614923515469790777, 7.985509700838747992914312518141, 8.253212365929126362388610978333, 9.350671456009366050953637904590

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