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Properties

Label 2-1040-13.4-c1-0-17
Degree $2$
Conductor $1040$
Sign $0.603 + 0.797i$
Analytic cond. $8.30444$
Root an. cond. $2.88174$
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  + (0.823 − 1.42i)3-s + i·5-s + (−1.33 + 0.769i)7-s + (0.143 + 0.249i)9-s + (−2.55 − 1.47i)11-s + (1.44 − 3.30i)13-s + (1.42 + 0.823i)15-s + (2.39 + 4.15i)17-s + (5.26 − 3.03i)19-s + 2.53i·21-s + (4.15 − 7.19i)23-s − 25-s + 5.41·27-s + (4.30 − 7.45i)29-s − 6.96i·31-s + ⋯
L(s)  = 1  + (0.475 − 0.823i)3-s + 0.447i·5-s + (−0.503 + 0.290i)7-s + (0.0479 + 0.0830i)9-s + (−0.769 − 0.444i)11-s + (0.400 − 0.916i)13-s + (0.368 + 0.212i)15-s + (0.581 + 1.00i)17-s + (1.20 − 0.696i)19-s + 0.552i·21-s + (0.865 − 1.49i)23-s − 0.200·25-s + 1.04·27-s + (0.798 − 1.38i)29-s − 1.25i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 + 0.797i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.603 + 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $0.603 + 0.797i$
Analytic conductor: \(8.30444\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1040} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1040,\ (\ :1/2),\ 0.603 + 0.797i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.835767747\)
\(L(\frac12)\) \(\approx\) \(1.835767747\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
13 \( 1 + (-1.44 + 3.30i)T \)
good3 \( 1 + (-0.823 + 1.42i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (1.33 - 0.769i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.55 + 1.47i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.39 - 4.15i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.26 + 3.03i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.15 + 7.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.30 + 7.45i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.96iT - 31T^{2} \)
37 \( 1 + (3.70 + 2.13i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-10.2 - 5.89i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.53 - 4.39i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 6.64iT - 47T^{2} \)
53 \( 1 + 5.71T + 53T^{2} \)
59 \( 1 + (-1.08 + 0.625i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.24 - 9.09i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.25 + 5.34i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (8.09 - 4.67i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 10.1iT - 73T^{2} \)
79 \( 1 + 7.47T + 79T^{2} \)
83 \( 1 - 6.44iT - 83T^{2} \)
89 \( 1 + (-2.70 - 1.56i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.91 - 2.25i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.862309516406413232024259346861, −8.810072022344531085278630851011, −7.891497665064487096056529630939, −7.59698438925700330427331480189, −6.36025547309867559316366203141, −5.81124840983634955946547196718, −4.51671405802203863549195870918, −2.99507574381760971789940712995, −2.63138767545204172877636432901, −0.928143651124725747107566908124, 1.30687765496368537700302505010, 3.07202396979428010746838710682, 3.66981336416594453845599616401, 4.84802042484823642370386925152, 5.44972075948912769056866700933, 6.91122089995255767863230824451, 7.45561916606623568577055007099, 8.713270930170475743098890479352, 9.273697666587740956186113902323, 9.917500754424143042643146326212

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