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Properties

Label 2-1040-13.12-c3-0-20
Degree $2$
Conductor $1040$
Sign $-0.330 - 0.943i$
Analytic cond. $61.3619$
Root an. cond. $7.83338$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7.17·3-s − 5i·5-s + 5.20i·7-s + 24.4·9-s + 46.2i·11-s + (−44.2 + 15.5i)13-s − 35.8i·15-s − 14.9·17-s + 75.6i·19-s + 37.3i·21-s + 134.·23-s − 25·25-s − 18.2·27-s − 236.·29-s − 65.0i·31-s + ⋯
L(s)  = 1  + 1.38·3-s − 0.447i·5-s + 0.281i·7-s + 0.905·9-s + 1.26i·11-s + (−0.943 + 0.330i)13-s − 0.617i·15-s − 0.212·17-s + 0.914i·19-s + 0.388i·21-s + 1.22·23-s − 0.200·25-s − 0.130·27-s − 1.51·29-s − 0.377i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.330 - 0.943i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.330 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $-0.330 - 0.943i$
Analytic conductor: \(61.3619\)
Root analytic conductor: \(7.83338\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1040} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1040,\ (\ :3/2),\ -0.330 - 0.943i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.167483132\)
\(L(\frac12)\) \(\approx\) \(2.167483132\)
\(L(\frac{5}{2})\) 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 \( 1 + 5iT \)
13 \( 1 + (44.2 - 15.5i)T \)
good3 \( 1 - 7.17T + 27T^{2} \)
7 \( 1 - 5.20iT - 343T^{2} \)
11 \( 1 - 46.2iT - 1.33e3T^{2} \)
17 \( 1 + 14.9T + 4.91e3T^{2} \)
19 \( 1 - 75.6iT - 6.85e3T^{2} \)
23 \( 1 - 134.T + 1.21e4T^{2} \)
29 \( 1 + 236.T + 2.43e4T^{2} \)
31 \( 1 + 65.0iT - 2.97e4T^{2} \)
37 \( 1 - 311. iT - 5.06e4T^{2} \)
41 \( 1 - 86.3iT - 6.89e4T^{2} \)
43 \( 1 + 299.T + 7.95e4T^{2} \)
47 \( 1 + 108. iT - 1.03e5T^{2} \)
53 \( 1 + 700.T + 1.48e5T^{2} \)
59 \( 1 - 287. iT - 2.05e5T^{2} \)
61 \( 1 - 59.0T + 2.26e5T^{2} \)
67 \( 1 - 304. iT - 3.00e5T^{2} \)
71 \( 1 + 274. iT - 3.57e5T^{2} \)
73 \( 1 + 835. iT - 3.89e5T^{2} \)
79 \( 1 - 382.T + 4.93e5T^{2} \)
83 \( 1 - 1.49e3iT - 5.71e5T^{2} \)
89 \( 1 - 1.49e3iT - 7.04e5T^{2} \)
97 \( 1 - 1.50e3iT - 9.12e5T^{2} \)
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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

−9.470538279652245374622595703305, −9.148761510109976103960552734598, −8.109533715898269603013185274145, −7.55255948071751414948683106316, −6.67292594697428125069322786663, −5.27416613512727568660535276321, −4.46597566564157970012658240749, −3.44134438559212823191778816234, −2.37087902325520208484151157882, −1.61644217281294846266792301925, 0.39465858951455778273665959220, 2.02835502811516819222745925984, 3.01604790978399985046725205598, 3.52830220530309704727577416500, 4.80970312130971383307294722311, 5.91039673475899547199730485488, 7.14810109875135845069076102772, 7.57040324850839352304017701257, 8.631172452623738283602466500644, 9.093157823519825174125943886401

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