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Properties

Label 2-1078-7.2-c1-0-21
Degree $2$
Conductor $1078$
Sign $0.701 + 0.712i$
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 − 1.73i)5-s + 0.999·8-s + (1.5 − 2.59i)9-s + (0.999 + 1.73i)10-s + (0.5 + 0.866i)11-s − 2·13-s + (−0.5 + 0.866i)16-s + (1 + 1.73i)17-s + (1.5 + 2.59i)18-s − 1.99·20-s − 0.999·22-s + (4 − 6.92i)23-s + (0.500 + 0.866i)25-s + (1 − 1.73i)26-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.447 − 0.774i)5-s + 0.353·8-s + (0.5 − 0.866i)9-s + (0.316 + 0.547i)10-s + (0.150 + 0.261i)11-s − 0.554·13-s + (−0.125 + 0.216i)16-s + (0.242 + 0.420i)17-s + (0.353 + 0.612i)18-s − 0.447·20-s − 0.213·22-s + (0.834 − 1.44i)23-s + (0.100 + 0.173i)25-s + (0.196 − 0.339i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $0.701 + 0.712i$
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: \((2,\ 1078,\ (\ :1/2),\ 0.701 + 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.326344722\)
\(L(\frac12)\) \(\approx\) \(1.326344722\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4 + 6.92i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + (4 + 6.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1 + 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (-4 + 6.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
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 \( 1 - 16T + 71T^{2} \)
73 \( 1 + (7 + 12.1i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−9.576212670992838019137758205844, −9.011698895514839308692975717180, −8.224681714176907893718533392516, −7.17583562772659924787213016686, −6.53410431425581507257422649546, −5.53233403050622131003540380692, −4.74636178244840463950561789917, −3.72149607587034049357444345602, −2.03070185263135945919286971450, −0.71035042443734153646141443209, 1.47292060864213751781595066973, 2.58231321597246310255879939346, 3.46369201794237708113870999252, 4.75785660731975300733394881715, 5.60637537270968964324739906742, 6.94123673901395241388060641287, 7.40719720049659547162404279098, 8.432652046097111286452680014232, 9.430386257736696122716414111611, 10.00070299831192740791114282593

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