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Properties

Label 2-1078-7.2-c1-0-23
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 + (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 + 4·13-s + 3.99·15-s + (−0.5 + 0.866i)16-s + (0.499 + 0.866i)18-s + (2 − 3.46i)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 + 1.10·13-s + 1.03·15-s + (−0.125 + 0.216i)16-s + (0.117 + 0.204i)18-s + (0.458 − 0.794i)19-s − 0.447·20-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

\(L(1)\) \(\approx\) \(2.581669927\)
\(L(\frac12)\) \(\approx\) \(2.581669927\)
\(L(\frac{3}{2})\) 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 \( 1 - 4T + 13T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + (5 + 8.66i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3 + 5.19i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (-5 + 8.66i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-7 - 12.1i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5 - 8.66i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4 + 6.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 + (-2 - 3.46i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (8 - 13.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 + (-5 + 8.66i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 6T + 97T^{2} \)
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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

−9.729518703746625427036580236451, −9.033285076876109913253813217025, −8.678046288002148475484347135518, −7.39213862833059705124936528491, −5.96189373856331395141331590868, −5.32632710212094652290359697579, −4.26311917124467220159868590220, −3.66539088643647559814240604776, −2.55196958815236995648887714010, −1.11876509928895296265418184908, 1.53096879039288438458703556727, 2.68805756499205694979615474203, 3.60901393889998111002004265296, 4.93056120075052739828509316368, 6.09765282408618968103522858111, 6.62899288222405712162424905873, 7.37919605047910310857874139410, 8.202460298631693284486675252214, 8.769125756712090139796204094030, 10.01350204916070635451877289179

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