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Properties

Label 2-2320-145.144-c1-0-38
Degree $2$
Conductor $2320$
Sign $0.889 - 0.456i$
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  − 0.367·3-s + (1.82 + 1.29i)5-s − 2.99i·7-s − 2.86·9-s + 1.08i·11-s − 1.37i·13-s + (−0.671 − 0.474i)15-s − 0.212·17-s + 6.70i·19-s + 1.10i·21-s + 1.06i·23-s + (1.67 + 4.71i)25-s + 2.15·27-s + (2.49 − 4.77i)29-s + 4.22i·31-s + ⋯
L(s)  = 1  − 0.212·3-s + (0.816 + 0.576i)5-s − 1.13i·7-s − 0.954·9-s + 0.328i·11-s − 0.381i·13-s + (−0.173 − 0.122i)15-s − 0.0514·17-s + 1.53i·19-s + 0.240i·21-s + 0.221i·23-s + (0.334 + 0.942i)25-s + 0.414·27-s + (0.463 − 0.885i)29-s + 0.759i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.889 - 0.456i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.889 - 0.456i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2320\)    =    \(2^{4} \cdot 5 \cdot 29\)
Sign: $0.889 - 0.456i$
Analytic conductor: \(18.5252\)
Root analytic conductor: \(4.30410\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2320} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2320,\ (\ :1/2),\ 0.889 - 0.456i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.730003332\)
\(L(\frac12)\) \(\approx\) \(1.730003332\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.82 - 1.29i)T \)
29 \( 1 + (-2.49 + 4.77i)T \)
good3 \( 1 + 0.367T + 3T^{2} \)
7 \( 1 + 2.99iT - 7T^{2} \)
11 \( 1 - 1.08iT - 11T^{2} \)
13 \( 1 + 1.37iT - 13T^{2} \)
17 \( 1 + 0.212T + 17T^{2} \)
19 \( 1 - 6.70iT - 19T^{2} \)
23 \( 1 - 1.06iT - 23T^{2} \)
31 \( 1 - 4.22iT - 31T^{2} \)
37 \( 1 - 5.72T + 37T^{2} \)
41 \( 1 - 10.3iT - 41T^{2} \)
43 \( 1 - 10.8T + 43T^{2} \)
47 \( 1 - 6.38T + 47T^{2} \)
53 \( 1 + 10.0iT - 53T^{2} \)
59 \( 1 - 5.28T + 59T^{2} \)
61 \( 1 - 3.38iT - 61T^{2} \)
67 \( 1 + 10.2iT - 67T^{2} \)
71 \( 1 - 5.04T + 71T^{2} \)
73 \( 1 + 3.51T + 73T^{2} \)
79 \( 1 + 8.85iT - 79T^{2} \)
83 \( 1 - 12.4iT - 83T^{2} \)
89 \( 1 - 2.95iT - 89T^{2} \)
97 \( 1 - 16.0T + 97T^{2} \)
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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

−9.203627102936649282237827748747, −8.103263324142196843406749900446, −7.57896380546999407766251411467, −6.57693085324072920006352990059, −6.02560859918687396833705145282, −5.24954181377289184927662779296, −4.16233682245354328695249624321, −3.25530328949253979997796390827, −2.28626412113608997454847022931, −1.00177674549965913199340154340, 0.74619309900459085431190112885, 2.31649144243449649184347188619, 2.74833289273959399710371442345, 4.27328634480495406295514818932, 5.21223588407278888068685388512, 5.74410358434502688949487414804, 6.34161504927994886190258565720, 7.36600889335346990520643877611, 8.570072937891955611828954518687, 8.941891498751265960253099783054

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