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Properties

Label 2-1078-7.4-c1-0-6
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 + (−1.20 + 2.09i)3-s + (−0.499 + 0.866i)4-s + (−0.292 − 0.507i)5-s + 2.41·6-s + 0.999·8-s + (−1.41 − 2.44i)9-s + (−0.292 + 0.507i)10-s + (−0.5 + 0.866i)11-s + (−1.20 − 2.09i)12-s + 3.82·13-s + 1.41·15-s + (−0.5 − 0.866i)16-s + (1.82 − 3.16i)17-s + (−1.41 + 2.44i)18-s + (−0.292 − 0.507i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.696 + 1.20i)3-s + (−0.249 + 0.433i)4-s + (−0.130 − 0.226i)5-s + 0.985·6-s + 0.353·8-s + (−0.471 − 0.816i)9-s + (−0.0926 + 0.160i)10-s + (−0.150 + 0.261i)11-s + (−0.348 − 0.603i)12-s + 1.06·13-s + 0.365·15-s + (−0.125 − 0.216i)16-s + (0.443 − 0.768i)17-s + (−0.333 + 0.577i)18-s + (−0.0671 − 0.116i)19-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: $\chi_{1078} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1078,\ (\ :1/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9278820641\)
\(L(\frac12)\) \(\approx\) \(0.9278820641\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.20 - 2.09i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.292 + 0.507i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 - 3.82T + 13T^{2} \)
17 \( 1 + (-1.82 + 3.16i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.292 + 0.507i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.12 - 5.40i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.65T + 29T^{2} \)
31 \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.70 - 8.15i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 5.41T + 41T^{2} \)
43 \( 1 + 5.65T + 43T^{2} \)
47 \( 1 + (5.24 + 9.08i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.94 - 6.84i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.79 - 4.83i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.91 - 10.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.37 - 2.38i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.0T + 71T^{2} \)
73 \( 1 + (4.70 - 8.15i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.62 - 11.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 12.1T + 83T^{2} \)
89 \( 1 + (-6.24 - 10.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 3.82T + 97T^{2} \)
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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

−10.17954017375974502192802743617, −9.399526193547921689288861321950, −8.716234895246902383226557494032, −7.75098543146084733288471574219, −6.59433945383640406499528540087, −5.44564494912569839209184887590, −4.75686180566901043375997522135, −3.87862954999686439472186788093, −2.91865026607353780100327605550, −1.13214929686437944892638840308, 0.63712503260384284201440131060, 1.78676092649032180415758147830, 3.40756788457424491678596021631, 4.79049274183026592033559907792, 6.03924451627818579422295235550, 6.21584839128916204335007498033, 7.20918272536583867141327357672, 7.896595722433195307927465709286, 8.625613493641034964049706320499, 9.599912921802389419726505640466

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