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Properties

Label 2-1040-13.12-c3-0-31
Degree $2$
Conductor $1040$
Sign $0.617 - 0.786i$
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  + 3.08·3-s − 5i·5-s + 17.4i·7-s − 17.4·9-s − 3.40i·11-s + (−36.8 − 28.9i)13-s − 15.4i·15-s + 102.·17-s + 4.59i·19-s + 53.7i·21-s − 32.4·23-s − 25·25-s − 137.·27-s + 241.·29-s − 124. i·31-s + ⋯
L(s)  = 1  + 0.593·3-s − 0.447i·5-s + 0.941i·7-s − 0.647·9-s − 0.0933i·11-s + (−0.786 − 0.617i)13-s − 0.265i·15-s + 1.45·17-s + 0.0555i·19-s + 0.558i·21-s − 0.293·23-s − 0.200·25-s − 0.977·27-s + 1.54·29-s − 0.723i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $0.617 - 0.786i$
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: \((2,\ 1040,\ (\ :3/2),\ 0.617 - 0.786i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.113395482\)
\(L(\frac12)\) \(\approx\) \(2.113395482\)
\(L(\frac{5}{2})\) 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 \( 1 + (36.8 + 28.9i)T \)
good3 \( 1 - 3.08T + 27T^{2} \)
7 \( 1 - 17.4iT - 343T^{2} \)
11 \( 1 + 3.40iT - 1.33e3T^{2} \)
17 \( 1 - 102.T + 4.91e3T^{2} \)
19 \( 1 - 4.59iT - 6.85e3T^{2} \)
23 \( 1 + 32.4T + 1.21e4T^{2} \)
29 \( 1 - 241.T + 2.43e4T^{2} \)
31 \( 1 + 124. iT - 2.97e4T^{2} \)
37 \( 1 - 184. iT - 5.06e4T^{2} \)
41 \( 1 - 377. iT - 6.89e4T^{2} \)
43 \( 1 - 297.T + 7.95e4T^{2} \)
47 \( 1 - 292. iT - 1.03e5T^{2} \)
53 \( 1 - 371.T + 1.48e5T^{2} \)
59 \( 1 - 358. iT - 2.05e5T^{2} \)
61 \( 1 + 323.T + 2.26e5T^{2} \)
67 \( 1 - 795. iT - 3.00e5T^{2} \)
71 \( 1 - 909. iT - 3.57e5T^{2} \)
73 \( 1 - 526. iT - 3.89e5T^{2} \)
79 \( 1 + 688.T + 4.93e5T^{2} \)
83 \( 1 + 1.14e3iT - 5.71e5T^{2} \)
89 \( 1 - 1.20e3iT - 7.04e5T^{2} \)
97 \( 1 + 695. iT - 9.12e5T^{2} \)
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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

−9.648535090279056662100312068633, −8.683558459549579544997860193640, −8.193457527563985431785854516661, −7.42779849555500562349499060423, −5.98243204453770700979511343170, −5.51187818027141522143875757188, −4.42981028495313225843207101890, −3.05847468034527081609981226040, −2.52821645832580888196274449717, −1.00948478214256148341077242797, 0.55547037034181353325476315089, 2.05671709010264730300372370407, 3.10728580430300015276790067407, 3.90858766834721537058519248193, 5.03153626420312112211192031956, 6.08750951488571219170195979562, 7.15760553137750715411200566943, 7.64005544024802345056025317248, 8.584391150210314312158368177590, 9.448321959819539193356215201793

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