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Properties

Label 2-1078-7.4-c1-0-1
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 + (0.207 − 0.358i)3-s + (−0.499 + 0.866i)4-s + (−1.70 − 2.95i)5-s − 0.414·6-s + 0.999·8-s + (1.41 + 2.44i)9-s + (−1.70 + 2.95i)10-s + (−0.5 + 0.866i)11-s + (0.207 + 0.358i)12-s − 1.82·13-s − 1.41·15-s + (−0.5 − 0.866i)16-s + (−3.82 + 6.63i)17-s + (1.41 − 2.44i)18-s + (−1.70 − 2.95i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.119 − 0.207i)3-s + (−0.249 + 0.433i)4-s + (−0.763 − 1.32i)5-s − 0.169·6-s + 0.353·8-s + (0.471 + 0.816i)9-s + (−0.539 + 0.935i)10-s + (−0.150 + 0.261i)11-s + (0.0597 + 0.103i)12-s − 0.507·13-s − 0.365·15-s + (−0.125 − 0.216i)16-s + (−0.928 + 1.60i)17-s + (0.333 − 0.577i)18-s + (−0.391 − 0.678i)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

\(L(1)\) \(\approx\) \(0.3880771842\)
\(L(\frac12)\) \(\approx\) \(0.3880771842\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.207 + 0.358i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.70 + 2.95i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 + 1.82T + 13T^{2} \)
17 \( 1 + (3.82 - 6.63i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.70 + 2.95i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.12 + 1.94i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 8.65T + 29T^{2} \)
31 \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.29 - 5.70i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.58T + 41T^{2} \)
43 \( 1 - 5.65T + 43T^{2} \)
47 \( 1 + (-3.24 - 5.61i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.94 + 10.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.20 - 7.28i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.08 - 5.34i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.62 - 9.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.07T + 71T^{2} \)
73 \( 1 + (3.29 - 5.70i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.37 - 4.11i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 16.1T + 83T^{2} \)
89 \( 1 + (2.24 + 3.88i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 1.82T + 97T^{2} \)
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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

−10.07361611092635202333791061851, −9.033149705210644085282685086038, −8.499900757533011943406358786573, −7.80727125452774452770342300399, −7.00992106102348367850459635149, −5.54654420051932405104961532181, −4.45774813667395885563101136753, −4.11045201834538240984786495522, −2.42339818101168106066961177737, −1.40449394789152773808825172811, 0.19556023110908663204122553119, 2.38261911480719988017602144868, 3.56103390192155205347672395577, 4.34023566181138521046815865278, 5.67361227770173730523194917779, 6.58503505583875957714547337521, 7.37519485085291131560772692778, 7.68003058040779525782300595886, 9.083209414753065558805633933324, 9.515916960006289856075114318844

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