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Properties

Label 2-1078-7.4-c1-0-7
Degree $2$
Conductor $1078$
Sign $-0.900 - 0.435i$
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  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (1.41 + 2.44i)5-s − 0.999·8-s + (1.5 + 2.59i)9-s + (−1.41 + 2.44i)10-s + (−0.5 + 0.866i)11-s − 4.24·13-s + (−0.5 − 0.866i)16-s + (−1.41 + 2.44i)17-s + (−1.5 + 2.59i)18-s + (0.707 + 1.22i)19-s − 2.82·20-s − 0.999·22-s + (−3 − 5.19i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.632 + 1.09i)5-s − 0.353·8-s + (0.5 + 0.866i)9-s + (−0.447 + 0.774i)10-s + (−0.150 + 0.261i)11-s − 1.17·13-s + (−0.125 − 0.216i)16-s + (−0.342 + 0.594i)17-s + (−0.353 + 0.612i)18-s + (0.162 + 0.280i)19-s − 0.632·20-s − 0.213·22-s + (−0.625 − 1.08i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 - 0.435i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.900 - 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $-0.900 - 0.435i$
Analytic conductor: \(8.60787\)
Root analytic conductor: \(2.93391\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1078} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1078,\ (\ :1/2),\ -0.900 - 0.435i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.781419505\)
\(L(\frac12)\) \(\approx\) \(1.781419505\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 \)
11 \( 1 + (0.5 - 0.866i)T \)
good3 \( 1 + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.41 - 2.44i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 + 4.24T + 13T^{2} \)
17 \( 1 + (1.41 - 2.44i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.707 - 1.22i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 8T + 29T^{2} \)
31 \( 1 + (0.707 - 1.22i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3 - 5.19i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 8.48T + 41T^{2} \)
43 \( 1 - 10T + 43T^{2} \)
47 \( 1 + (3.53 + 6.12i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (7.07 - 12.2i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.12 + 3.67i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-4.24 + 7.34i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.07T + 83T^{2} \)
89 \( 1 + (-9.19 - 15.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 1.41T + 97T^{2} \)
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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

−10.37363611956408009205348906307, −9.541848728533306257612743801610, −8.316917306575182513391884798859, −7.60422103181112350605046160850, −6.76810549761453412032274550872, −6.21045957179983763022317803790, −5.05547631760692165679752550972, −4.35008787844682506633222158031, −2.90798021682824209640675722723, −2.10007316785247457113001795570, 0.69090053827920293141263638921, 1.91466849033878411543357018002, 3.12278555001894664536278834516, 4.36969082957107006493974598273, 5.02742531725320234416667921299, 5.88749475774885026814310202252, 6.90414035134818985801014931290, 7.981334382265610009495601292494, 9.129899054122288020744107854881, 9.505563028476838408901493146265

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