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Properties

Label 2-2300-460.367-c0-0-13
Degree $2$
Conductor $2300$
Sign $-0.973 + 0.229i$
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  + (−0.707 + 0.707i)2-s + (−1.41 − 1.41i)3-s − 1.00i·4-s + 2.00·6-s + (0.707 + 0.707i)8-s + 3.00i·9-s + (−1.41 + 1.41i)12-s − 1.00·16-s + (−2.12 − 2.12i)18-s + (−0.707 − 0.707i)23-s − 2.00i·24-s + (2.82 − 2.82i)27-s − 2i·29-s + (0.707 − 0.707i)32-s + 3.00·36-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (−1.41 − 1.41i)3-s − 1.00i·4-s + 2.00·6-s + (0.707 + 0.707i)8-s + 3.00i·9-s + (−1.41 + 1.41i)12-s − 1.00·16-s + (−2.12 − 2.12i)18-s + (−0.707 − 0.707i)23-s − 2.00i·24-s + (2.82 − 2.82i)27-s − 2i·29-s + (0.707 − 0.707i)32-s + 3.00·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2300\)    =    \(2^{2} \cdot 5^{2} \cdot 23\)
Sign: $-0.973 + 0.229i$
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: \((2,\ 2300,\ (\ :0),\ -0.973 + 0.229i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1724705734\)
\(L(\frac12)\) \(\approx\) \(0.1724705734\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 \)
23 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 - T^{2} \)
29 \( 1 + 2iT - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + 2T + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - iT^{2} \)
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

−8.299233342705119670813739282790, −8.055651722830062936910617886791, −7.16167101327055255850819678090, −6.51509640499890077198071800474, −6.04366910315545801900415012159, −5.25879294326373930697219375104, −4.48288889606064715960729081146, −2.35585895802950560395774159947, −1.47212383855488152899696764243, −0.19226412830569124651407128144, 1.43631457912714778588903993710, 3.23151578741895237170178129096, 3.77871023601432759223989320643, 4.77440598202161769190607169355, 5.37650258454064254584537778090, 6.45549004575628318765914825492, 7.11412836527133099617508142819, 8.398454744861584548399529049598, 9.078265243391217089628225127070, 9.818969016751791271621907490846

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