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Properties

Label 2-2320-5.4-c1-0-21
Degree $2$
Conductor $2320$
Sign $-0.973 + 0.228i$
Analytic cond. $18.5252$
Root an. cond. $4.30410$
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  + 2.89i·3-s + (2.17 − 0.511i)5-s + 3.91i·7-s − 5.35·9-s − 2.65·11-s + 5.62i·13-s + (1.47 + 6.29i)15-s + 1.86i·17-s + 1.69·19-s − 11.3·21-s − 0.691i·23-s + (4.47 − 2.22i)25-s − 6.80i·27-s + 29-s + 0.654·31-s + ⋯
L(s)  = 1  + 1.66i·3-s + (0.973 − 0.228i)5-s + 1.47i·7-s − 1.78·9-s − 0.800·11-s + 1.56i·13-s + (0.381 + 1.62i)15-s + 0.453i·17-s + 0.389·19-s − 2.46·21-s − 0.144i·23-s + (0.895 − 0.445i)25-s − 1.30i·27-s + 0.185·29-s + 0.117·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.228i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.973 + 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2320\)    =    \(2^{4} \cdot 5 \cdot 29\)
Sign: $-0.973 + 0.228i$
Analytic conductor: \(18.5252\)
Root analytic conductor: \(4.30410\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2320} (929, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2320,\ (\ :1/2),\ -0.973 + 0.228i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.710413418\)
\(L(\frac12)\) \(\approx\) \(1.710413418\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.17 + 0.511i)T \)
29 \( 1 - T \)
good3 \( 1 - 2.89iT - 3T^{2} \)
7 \( 1 - 3.91iT - 7T^{2} \)
11 \( 1 + 2.65T + 11T^{2} \)
13 \( 1 - 5.62iT - 13T^{2} \)
17 \( 1 - 1.86iT - 17T^{2} \)
19 \( 1 - 1.69T + 19T^{2} \)
23 \( 1 + 0.691iT - 23T^{2} \)
31 \( 1 - 0.654T + 31T^{2} \)
37 \( 1 - 3.91iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 10.7iT - 43T^{2} \)
47 \( 1 + 4.93iT - 47T^{2} \)
53 \( 1 + 7.67iT - 53T^{2} \)
59 \( 1 - 10T + 59T^{2} \)
61 \( 1 + 4.70T + 61T^{2} \)
67 \( 1 - 6.47iT - 67T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 - 10.5iT - 73T^{2} \)
79 \( 1 + 2.05T + 79T^{2} \)
83 \( 1 - 1.86iT - 83T^{2} \)
89 \( 1 + 3.30T + 89T^{2} \)
97 \( 1 - 0.384iT - 97T^{2} \)
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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

−9.504850002368322318772181141834, −8.698237179855685297355820358847, −8.529542432307171208496799042352, −6.88413393201920117935761335993, −5.93624621604132127520501866610, −5.34332235849906814869407356583, −4.81687547038270336647311920195, −3.83847776055883172505279563661, −2.72545047124166660629465658484, −1.99009805035840875910145322615, 0.58113630729285540521022343640, 1.36728790658117688890252510047, 2.56817512187527357391883841394, 3.23196180081241640343188802368, 4.79972494532517809752960647897, 5.71316966314357644580340185039, 6.31905658623011105237607097622, 7.24069881397528001771964997687, 7.59698828421938343505388702082, 8.201680598963021972704999026399

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