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Properties

Label 2-2300-115.68-c1-0-21
Degree $2$
Conductor $2300$
Sign $0.973 + 0.229i$
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  + (3.06 + 3.06i)7-s − 3i·9-s + (−2.41 − 2.41i)17-s + (3.39 − 3.39i)23-s − 8.78i·29-s + 10.7·31-s + (8.54 + 8.54i)37-s − 12.7·41-s + (6.78 − 6.78i)43-s + 11.7i·49-s + (3.71 − 3.71i)53-s + 14.7i·59-s + (9.19 − 9.19i)63-s + (−7.88 − 7.88i)67-s − 2.78·71-s + ⋯
L(s)  = 1  + (1.15 + 1.15i)7-s − i·9-s + (−0.585 − 0.585i)17-s + (0.707 − 0.707i)23-s − 1.63i·29-s + 1.93·31-s + (1.40 + 1.40i)37-s − 1.99·41-s + (1.03 − 1.03i)43-s + 1.68i·49-s + (0.510 − 0.510i)53-s + 1.92i·59-s + (1.15 − 1.15i)63-s + (−0.963 − 0.963i)67-s − 0.330·71-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2300\)    =    \(2^{2} \cdot 5^{2} \cdot 23\)
Sign: $0.973 + 0.229i$
Analytic conductor: \(18.3655\)
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: \((2,\ 2300,\ (\ :1/2),\ 0.973 + 0.229i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.077827902\)
\(L(\frac12)\) \(\approx\) \(2.077827902\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.39 + 3.39i)T \)
good3 \( 1 + 3iT^{2} \)
7 \( 1 + (-3.06 - 3.06i)T + 7iT^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + 13iT^{2} \)
17 \( 1 + (2.41 + 2.41i)T + 17iT^{2} \)
19 \( 1 + 19T^{2} \)
29 \( 1 + 8.78iT - 29T^{2} \)
31 \( 1 - 10.7T + 31T^{2} \)
37 \( 1 + (-8.54 - 8.54i)T + 37iT^{2} \)
41 \( 1 + 12.7T + 41T^{2} \)
43 \( 1 + (-6.78 + 6.78i)T - 43iT^{2} \)
47 \( 1 - 47iT^{2} \)
53 \( 1 + (-3.71 + 3.71i)T - 53iT^{2} \)
59 \( 1 - 14.7iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + (7.88 + 7.88i)T + 67iT^{2} \)
71 \( 1 + 2.78T + 71T^{2} \)
73 \( 1 + 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (1.75 - 1.75i)T - 83iT^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (-13.5 - 13.5i)T + 97iT^{2} \)
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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

−8.754338779701365307878653147742, −8.471057980555058891794738377480, −7.51733905310730625447797976784, −6.49234705166689687757963319509, −5.94487895324228125913949471960, −4.88274433721664469086349838352, −4.36306304973962012727960066215, −2.97667707836447661824522861403, −2.23306134107445438278650021336, −0.880525805034800126468020520888, 1.10181787025401780076528474952, 2.03773049338645973483799260333, 3.30002836744224767848141804817, 4.49501639058139248539325432142, 4.77329547646529945922922094827, 5.83351537440876876770906315834, 6.94126554018708869252923659638, 7.52967072635353854224158349199, 8.190180996644840100541246065651, 8.843899153196286790446197251816

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