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Properties

Label 2-2300-115.68-c1-0-4
Degree $2$
Conductor $2300$
Sign $0.331 - 0.943i$
Analytic cond. $18.3655$
Root an. cond. $4.28550$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.854 − 0.854i)3-s + (3.24 + 3.24i)7-s − 1.53i·9-s + 4.86i·11-s + (−4.90 − 4.90i)13-s + (3.33 + 3.33i)17-s + 1.13·19-s − 5.54i·21-s + (1.49 + 4.55i)23-s + (−3.87 + 3.87i)27-s + 0.233i·29-s + 6.17·31-s + (4.15 − 4.15i)33-s + (−5.16 − 5.16i)37-s + 8.39i·39-s + ⋯
L(s)  = 1  + (−0.493 − 0.493i)3-s + (1.22 + 1.22i)7-s − 0.512i·9-s + 1.46i·11-s + (−1.36 − 1.36i)13-s + (0.808 + 0.808i)17-s + 0.260·19-s − 1.20i·21-s + (0.311 + 0.950i)23-s + (−0.746 + 0.746i)27-s + 0.0432i·29-s + 1.10·31-s + (0.724 − 0.724i)33-s + (−0.849 − 0.849i)37-s + 1.34i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.331 - 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.331 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2300\)    =    \(2^{2} \cdot 5^{2} \cdot 23\)
Sign: $0.331 - 0.943i$
Analytic conductor: \(18.3655\)
Root analytic conductor: \(4.28550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2300} (1793, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2300,\ (\ :1/2),\ 0.331 - 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.347163844\)
\(L(\frac12)\) \(\approx\) \(1.347163844\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 + (-1.49 - 4.55i)T \)
good3 \( 1 + (0.854 + 0.854i)T + 3iT^{2} \)
7 \( 1 + (-3.24 - 3.24i)T + 7iT^{2} \)
11 \( 1 - 4.86iT - 11T^{2} \)
13 \( 1 + (4.90 + 4.90i)T + 13iT^{2} \)
17 \( 1 + (-3.33 - 3.33i)T + 17iT^{2} \)
19 \( 1 - 1.13T + 19T^{2} \)
29 \( 1 - 0.233iT - 29T^{2} \)
31 \( 1 - 6.17T + 31T^{2} \)
37 \( 1 + (5.16 + 5.16i)T + 37iT^{2} \)
41 \( 1 - 0.771T + 41T^{2} \)
43 \( 1 + (5.83 - 5.83i)T - 43iT^{2} \)
47 \( 1 + (6.70 - 6.70i)T - 47iT^{2} \)
53 \( 1 + (0.739 - 0.739i)T - 53iT^{2} \)
59 \( 1 + 5.83iT - 59T^{2} \)
61 \( 1 - 5.08iT - 61T^{2} \)
67 \( 1 + (1.97 + 1.97i)T + 67iT^{2} \)
71 \( 1 - 9.81T + 71T^{2} \)
73 \( 1 + (1.99 + 1.99i)T + 73iT^{2} \)
79 \( 1 + 9.05T + 79T^{2} \)
83 \( 1 + (7.84 - 7.84i)T - 83iT^{2} \)
89 \( 1 - 17.6T + 89T^{2} \)
97 \( 1 + (-10.8 - 10.8i)T + 97iT^{2} \)
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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

−9.270761021215067411879154755238, −8.075414130223851914512489311299, −7.75836811005305537643707825611, −6.90601804431501004402179609168, −5.86808237248862793098249683170, −5.24662208411943385736587530133, −4.70047797079374829462118108430, −3.25151010475211138829005603375, −2.19201829677159605521712353745, −1.30083290007116303897179815156, 0.51930225862526430273056973593, 1.82063694911297631051989491009, 3.11737889948966640858610648902, 4.29507192934476637387434563553, 4.82872700994743628149206390968, 5.39723378491221876970818495647, 6.65234171950172077561902518542, 7.30565735898907270957443902487, 8.096954847476289720566863476516, 8.727214804941093077124254538998

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