Properties

Label 2-1078-77.10-c1-0-12
Degree $2$
Conductor $1078$
Sign $0.862 - 0.506i$
Analytic cond. $8.60787$
Root an. cond. $2.93391$
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  + (−0.866 + 0.5i)2-s + (−1.60 − 0.923i)3-s + (0.499 − 0.866i)4-s + (2.80 − 1.61i)5-s + 1.84·6-s + 0.999i·8-s + (0.207 + 0.358i)9-s + (−1.61 + 2.80i)10-s + (−1.23 + 3.07i)11-s + (−1.60 + 0.923i)12-s + 6.10·13-s − 5.98·15-s + (−0.5 − 0.866i)16-s + (−4.06 + 7.04i)17-s + (−0.358 − 0.207i)18-s + (3.30 + 5.72i)19-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.923 − 0.533i)3-s + (0.249 − 0.433i)4-s + (1.25 − 0.723i)5-s + 0.754·6-s + 0.353i·8-s + (0.0690 + 0.119i)9-s + (−0.511 + 0.886i)10-s + (−0.372 + 0.927i)11-s + (−0.461 + 0.266i)12-s + 1.69·13-s − 1.54·15-s + (−0.125 − 0.216i)16-s + (−0.986 + 1.70i)17-s + (−0.0845 − 0.0488i)18-s + (0.758 + 1.31i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.862 - 0.506i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.862 - 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $0.862 - 0.506i$
Analytic conductor: \(8.60787\)
Root analytic conductor: \(2.93391\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1078} (1011, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1078,\ (\ :1/2),\ 0.862 - 0.506i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.041840275\)
\(L(\frac12)\) \(\approx\) \(1.041840275\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 \)
11 \( 1 + (1.23 - 3.07i)T \)
good3 \( 1 + (1.60 + 0.923i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-2.80 + 1.61i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 - 6.10T + 13T^{2} \)
17 \( 1 + (4.06 - 7.04i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.30 - 5.72i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.65 - 4.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.32iT - 29T^{2} \)
31 \( 1 + (2.03 + 1.17i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.64 + 4.57i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 1.22T + 41T^{2} \)
43 \( 1 - 3.73iT - 43T^{2} \)
47 \( 1 + (-1.47 + 0.853i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.99 + 3.44i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-7.82 - 4.51i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.27 - 7.41i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.17 - 2.03i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.17T + 71T^{2} \)
73 \( 1 + (-5.49 + 9.51i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.34 - 4.24i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.60T + 83T^{2} \)
89 \( 1 + (4.03 - 2.33i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.82iT - 97T^{2} \)
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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

−9.945283375113309659274069231474, −9.052155694281030951674123007131, −8.446295763730322380435090802783, −7.36201204141011888478993485048, −6.36037345506110655769445950145, −5.85534996054872131324809834013, −5.32192734439695317018514447321, −3.86499154449014560000855735699, −1.83225686606577345012204487741, −1.27955709671131543153445704834, 0.72246074377781064028096468838, 2.40610177342955345083362651917, 3.23030074322864856796942278070, 4.80257351484953701222490908275, 5.59373957520493202980367666869, 6.44246177176939832458737628876, 7.05234139690774081055061793625, 8.551949862628414365705018844267, 9.087263196019856241479650942082, 10.00505866388646882369677239201

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