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Properties

Label 2-1078-7.4-c1-0-17
Degree $2$
Conductor $1078$
Sign $0.605 + 0.795i$
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 + (1 − 1.73i)3-s + (−0.499 + 0.866i)4-s + (1 + 1.73i)5-s − 1.99·6-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (0.999 − 1.73i)10-s + (−0.5 + 0.866i)11-s + (0.999 + 1.73i)12-s + 2·13-s + 3.99·15-s + (−0.5 − 0.866i)16-s + (−0.499 + 0.866i)18-s + (1 + 1.73i)19-s − 1.99·20-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.577 − 0.999i)3-s + (−0.249 + 0.433i)4-s + (0.447 + 0.774i)5-s − 0.816·6-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.316 − 0.547i)10-s + (−0.150 + 0.261i)11-s + (0.288 + 0.499i)12-s + 0.554·13-s + 1.03·15-s + (−0.125 − 0.216i)16-s + (−0.117 + 0.204i)18-s + (0.229 + 0.397i)19-s − 0.447·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $0.605 + 0.795i$
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: \((2,\ 1078,\ (\ :1/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.825907429\)
\(L(\frac12)\) \(\approx\) \(1.825907429\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 1.73i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
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 \( 1 - 8T + 41T^{2} \)
43 \( 1 - 12T + 43T^{2} \)
47 \( 1 + (6 + 10.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1 + 1.73i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5 - 8.66i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6 + 10.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4T + 71T^{2} \)
73 \( 1 + (6 - 10.3i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 18T + 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 12T + 97T^{2} \)
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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

−9.878480197869522918873785902868, −8.833126940553592347636614710483, −8.185673751954384129227300022030, −7.29904018668731221920527153075, −6.70386142176735825779759230728, −5.63999097904389062575008372897, −4.21956228880155036300121981627, −2.98330363049331644095271697287, −2.30660920713042264604131737042, −1.21874606892508270594140318378, 1.10051110655016347381152921607, 2.79555753753060316965791167108, 4.02676480271910917857165975411, 4.80590847402103469607571089897, 5.68212035210045430791930913610, 6.58148929550446217701169107224, 7.78695518664863755018312269860, 8.528670673808767553411287838081, 9.277313174607398447291313851018, 9.569132257440736583791078470820

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