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Properties

Label 2-1078-77.54-c1-0-36
Degree $2$
Conductor $1078$
Sign $0.325 + 0.945i$
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 + (−2.04 − 2.60i)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 + (−0.617 − 0.786i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.325 + 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.325 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $0.325 + 0.945i$
Analytic conductor: \(8.60787\)
Root analytic conductor: \(2.93391\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1078} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1078,\ (\ :1/2),\ 0.325 + 0.945i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.305134054\)
\(L(\frac12)\) \(\approx\) \(2.305134054\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.04 + 2.60i)T \)
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

−9.187763066696678451900366339668, −8.597273633498227712953109464138, −8.133435526651461663562606784315, −7.32053603189771507407690203250, −6.53735479138092404667959982935, −5.19062239057202807343025890948, −4.49931987296736069789618636712, −3.33700142314197226470292661570, −2.68555863164704608171677718251, −0.77269130955830644649792990444, 1.81143762114367535927371728006, 3.27736152786095064222703183077, 3.64157692891607798252249031345, 4.32068931045697770198065183825, 5.72747763506985006627470183824, 6.67543060669812766691711311560, 7.72881510166483725072429232484, 8.338065411408778771439385765692, 9.183376058238148155877397235485, 10.34715934733799197376026851922

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