Properties

Label 2-1078-77.10-c1-0-15
Degree $2$
Conductor $1078$
Sign $0.737 + 0.675i$
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 + (−1.35 − 0.779i)3-s + (0.499 − 0.866i)4-s + (−0.882 + 0.509i)5-s + 1.55·6-s + 0.999i·8-s + (−0.284 − 0.492i)9-s + (0.509 − 0.882i)10-s + (−0.510 + 3.27i)11-s + (−1.35 + 0.779i)12-s + 0.167·13-s + 1.58·15-s + (−0.5 − 0.866i)16-s + (1.47 − 2.55i)17-s + (0.492 + 0.284i)18-s + (0.155 + 0.269i)19-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.779 − 0.450i)3-s + (0.249 − 0.433i)4-s + (−0.394 + 0.227i)5-s + 0.636·6-s + 0.353i·8-s + (−0.0947 − 0.164i)9-s + (0.161 − 0.279i)10-s + (−0.153 + 0.988i)11-s + (−0.389 + 0.225i)12-s + 0.0463·13-s + 0.410·15-s + (−0.125 − 0.216i)16-s + (0.358 − 0.620i)17-s + (0.116 + 0.0670i)18-s + (0.0357 + 0.0619i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.737 + 0.675i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.737 + 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $0.737 + 0.675i$
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: \((2,\ 1078,\ (\ :1/2),\ 0.737 + 0.675i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6390889275\)
\(L(\frac12)\) \(\approx\) \(0.6390889275\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.510 - 3.27i)T \)
good3 \( 1 + (1.35 + 0.779i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.882 - 0.509i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 - 0.167T + 13T^{2} \)
17 \( 1 + (-1.47 + 2.55i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.155 - 0.269i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.237 - 0.411i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.89iT - 29T^{2} \)
31 \( 1 + (-2.20 - 1.27i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.04 + 5.27i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 10.2T + 41T^{2} \)
43 \( 1 + 10.1iT - 43T^{2} \)
47 \( 1 + (-3.28 + 1.89i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.21 - 7.29i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.40 + 3.12i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.93 + 10.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.19 + 8.99i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 14.5T + 71T^{2} \)
73 \( 1 + (-4.85 + 8.40i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.06 + 3.50i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 14.6T + 83T^{2} \)
89 \( 1 + (5.97 - 3.45i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 11.0iT - 97T^{2} \)
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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

−9.626854691442960638791594701519, −9.100621119346911285056521207040, −7.81981121975661663873249790242, −7.34043322134695179625843899168, −6.55853111221834544083259979645, −5.70328478825945639231257978908, −4.83270330269191814641526971141, −3.48314989032694408959816093009, −2.00656896824602390526253713071, −0.53140600647998185395252741392, 0.904286900774247887509463967626, 2.59397420152795211239054213956, 3.77131912698459013240984035281, 4.73970496755812863440185161434, 5.76669971191117415518385360272, 6.48959667080888251209099350948, 7.88404878244249586384355841766, 8.237427430941526756225668519488, 9.249320109736040787259566394885, 10.14598355599211497682297287156

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