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Properties

Label 2-1078-77.10-c1-0-32
Degree $2$
Conductor $1078$
Sign $-0.0514 + 0.998i$
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  + (−0.866 + 0.5i)2-s + (0.937 + 0.541i)3-s + (0.499 − 0.866i)4-s + (0.937 − 0.541i)5-s − 1.08·6-s + 0.999i·8-s + (−0.914 − 1.58i)9-s + (−0.541 + 0.937i)10-s + (−1.89 − 2.72i)11-s + (0.937 − 0.541i)12-s − 2.29·13-s + 1.17·15-s + (−0.5 − 0.866i)16-s + (1.58 + 0.914i)18-s + (−2.77 − 4.80i)19-s − 1.08i·20-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.541 + 0.312i)3-s + (0.249 − 0.433i)4-s + (0.419 − 0.242i)5-s − 0.441·6-s + 0.353i·8-s + (−0.304 − 0.527i)9-s + (−0.171 + 0.296i)10-s + (−0.570 − 0.821i)11-s + (0.270 − 0.156i)12-s − 0.636·13-s + 0.302·15-s + (−0.125 − 0.216i)16-s + (0.373 + 0.215i)18-s + (−0.635 − 1.10i)19-s − 0.242i·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0514 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0514 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $-0.0514 + 0.998i$
Analytic conductor: \(8.60787\)
Root analytic conductor: \(2.93391\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1078} (1011, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1078,\ (\ :1/2),\ -0.0514 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8654291966\)
\(L(\frac12)\) \(\approx\) \(0.8654291966\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 \)
11 \( 1 + (1.89 + 2.72i)T \)
good3 \( 1 + (-0.937 - 0.541i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.937 + 0.541i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 + 2.29T + 13T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.77 + 4.80i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.12 + 5.40i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.75iT - 29T^{2} \)
31 \( 1 + (-1.21 - 0.699i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3 + 5.19i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.59T + 41T^{2} \)
43 \( 1 - 4.24iT - 43T^{2} \)
47 \( 1 + (-4.25 + 2.45i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.24 - 3.88i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.23 - 3.60i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.77 + 4.80i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.82 - 10.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + (-7.83 + 13.5i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.34 - 4.24i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 13.3T + 83T^{2} \)
89 \( 1 + (3.08 - 1.78i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 8.60iT - 97T^{2} \)
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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

−9.482507291522289217704558097796, −8.809665645600790402172416068087, −8.308098096668193215613056516862, −7.26588726858999368758761893457, −6.32794484322269465917333688817, −5.52596599347215053668524878717, −4.48211259203452885335292683339, −3.15779507494671197164728086044, −2.19625617155395455593433466144, −0.40188440317170008875271207415, 1.85676325324257860796248941592, 2.39181180997615674351052448800, 3.60598474348826338108486569310, 4.89432168937704874835495822940, 5.95200936773586433261747246700, 7.03932384549211119558733738135, 7.84668473362682297778776805145, 8.292864750279741843222647724264, 9.394621477391678598800732789719, 10.11095728491693983503297923500

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