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Properties

Label 2-1040-13.12-c3-0-30
Degree $2$
Conductor $1040$
Sign $0.523 - 0.851i$
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  + 0.936·3-s + 5i·5-s + 0.201i·7-s − 26.1·9-s − 10.4i·11-s + (39.9 + 24.5i)13-s + 4.68i·15-s − 54.7·17-s − 129. i·19-s + 0.189i·21-s + 175.·23-s − 25·25-s − 49.7·27-s + 107.·29-s − 22.2i·31-s + ⋯
L(s)  = 1  + 0.180·3-s + 0.447i·5-s + 0.0108i·7-s − 0.967·9-s − 0.285i·11-s + (0.851 + 0.523i)13-s + 0.0806i·15-s − 0.781·17-s − 1.55i·19-s + 0.00196i·21-s + 1.59·23-s − 0.200·25-s − 0.354·27-s + 0.685·29-s − 0.128i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.523 - 0.851i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.523 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $0.523 - 0.851i$
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: \((2,\ 1040,\ (\ :3/2),\ 0.523 - 0.851i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.827891931\)
\(L(\frac12)\) \(\approx\) \(1.827891931\)
\(L(\frac{5}{2})\) 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 \( 1 + (-39.9 - 24.5i)T \)
good3 \( 1 - 0.936T + 27T^{2} \)
7 \( 1 - 0.201iT - 343T^{2} \)
11 \( 1 + 10.4iT - 1.33e3T^{2} \)
17 \( 1 + 54.7T + 4.91e3T^{2} \)
19 \( 1 + 129. iT - 6.85e3T^{2} \)
23 \( 1 - 175.T + 1.21e4T^{2} \)
29 \( 1 - 107.T + 2.43e4T^{2} \)
31 \( 1 + 22.2iT - 2.97e4T^{2} \)
37 \( 1 - 205. iT - 5.06e4T^{2} \)
41 \( 1 - 285. iT - 6.89e4T^{2} \)
43 \( 1 + 321.T + 7.95e4T^{2} \)
47 \( 1 - 379. iT - 1.03e5T^{2} \)
53 \( 1 - 506.T + 1.48e5T^{2} \)
59 \( 1 + 678. iT - 2.05e5T^{2} \)
61 \( 1 + 186.T + 2.26e5T^{2} \)
67 \( 1 - 925. iT - 3.00e5T^{2} \)
71 \( 1 - 376. iT - 3.57e5T^{2} \)
73 \( 1 - 671. iT - 3.89e5T^{2} \)
79 \( 1 - 593.T + 4.93e5T^{2} \)
83 \( 1 - 425. iT - 5.71e5T^{2} \)
89 \( 1 - 141. iT - 7.04e5T^{2} \)
97 \( 1 + 551. iT - 9.12e5T^{2} \)
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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

−9.498875569190944544814917980313, −8.795652113428599096527441042332, −8.254525148699617552350866639109, −6.93010107435558664083924026601, −6.50307012960946603744989847167, −5.38055091987125738498553700281, −4.41947007121299152829199003384, −3.17869666545291807718638892721, −2.51719304131901111155359977053, −0.938661160290058629693018538242, 0.54176959106086511108269653307, 1.86019759240404352429123016649, 3.09615809298461086311570215918, 4.00638641430521761599351352984, 5.20614200880570406647850784605, 5.86802057374033769029851565977, 6.88582433822139472722737571319, 7.927984444169034932023391445616, 8.684907015943350186201310890453, 9.126331171267795965157479474792

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