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Properties

Label 2-1078-7.2-c1-0-20
Degree $2$
Conductor $1078$
Sign $0.386 + 0.922i$
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 + (0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + 0.999·6-s − 0.999·8-s + (1 − 1.73i)9-s + (0.5 + 0.866i)11-s + (0.499 − 0.866i)12-s + 13-s + (−0.5 + 0.866i)16-s + (−3 − 5.19i)17-s + (−1 − 1.73i)18-s + (1 − 1.73i)19-s + 0.999·22-s + (3 − 5.19i)23-s + (−0.499 − 0.866i)24-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + 0.408·6-s − 0.353·8-s + (0.333 − 0.577i)9-s + (0.150 + 0.261i)11-s + (0.144 − 0.249i)12-s + 0.277·13-s + (−0.125 + 0.216i)16-s + (−0.727 − 1.26i)17-s + (−0.235 − 0.408i)18-s + (0.229 − 0.397i)19-s + 0.213·22-s + (0.625 − 1.08i)23-s + (−0.102 − 0.176i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $0.386 + 0.922i$
Analytic conductor: \(8.60787\)
Root analytic conductor: \(2.93391\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1078} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1078,\ (\ :1/2),\ 0.386 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.127838218\)
\(L(\frac12)\) \(\approx\) \(2.127838218\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-2.5 - 4.33i)T^{2} \)
13 \( 1 - T + 13T^{2} \)
17 \( 1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1 + 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 9T + 29T^{2} \)
31 \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.5 + 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.5 + 9.52i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.5 + 9.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-1 - 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.5 - 4.33i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (9 - 15.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 13T + 97T^{2} \)
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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

−9.587159948076271020332804923471, −9.255661462487002235834745161795, −8.347297881760747658864483890905, −7.01948105860642191046275300789, −6.40874128885435513045069454441, −4.98380498282920744647575634358, −4.49004701694734968738406571790, −3.37129338803553429510955927136, −2.54536368458609811717449923553, −0.926571860457696678279022541406, 1.45698573322672188167741303031, 2.79874522381940100093547824940, 3.96281355148462004812862051896, 4.88251649371443125774688056225, 5.91832664086939168897531281991, 6.72806210687619591578597191674, 7.44747866639599614575568703753, 8.408241477529121046090652651866, 8.748428681230517740595970847040, 10.08415165423804370100830699077

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