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Properties

Label 2-1078-77.54-c1-0-27
Degree $2$
Conductor $1078$
Sign $0.251 + 0.967i$
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 + (−1.60 + 0.923i)3-s + (0.499 + 0.866i)4-s + (−2.14 − 1.23i)5-s − 1.84·6-s + 0.999i·8-s + (0.207 − 0.358i)9-s + (−1.23 − 2.14i)10-s + (−1.30 + 3.05i)11-s + (−1.60 − 0.923i)12-s + 1.96·13-s + 4.56·15-s + (−0.5 + 0.866i)16-s + (−2.82 − 4.89i)17-s + (0.358 − 0.207i)18-s + (−0.453 + 0.785i)19-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.923 + 0.533i)3-s + (0.249 + 0.433i)4-s + (−0.957 − 0.552i)5-s − 0.754·6-s + 0.353i·8-s + (0.0690 − 0.119i)9-s + (−0.390 − 0.676i)10-s + (−0.392 + 0.919i)11-s + (−0.461 − 0.266i)12-s + 0.544·13-s + 1.17·15-s + (−0.125 + 0.216i)16-s + (−0.685 − 1.18i)17-s + (0.0845 − 0.0488i)18-s + (−0.104 + 0.180i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.251 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.251 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $0.251 + 0.967i$
Analytic conductor: \(8.60787\)
Root analytic conductor: \(2.93391\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1078} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1078,\ (\ :1/2),\ 0.251 + 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5117719703\)
\(L(\frac12)\) \(\approx\) \(0.5117719703\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.30 - 3.05i)T \)
good3 \( 1 + (1.60 - 0.923i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (2.14 + 1.23i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 - 1.96T + 13T^{2} \)
17 \( 1 + (2.82 + 4.89i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.453 - 0.785i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.468 + 0.811i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.50iT - 29T^{2} \)
31 \( 1 + (4.08 - 2.35i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.81 + 8.33i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.93T + 41T^{2} \)
43 \( 1 + 6.80iT - 43T^{2} \)
47 \( 1 + (3.46 + 2.00i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.28 + 5.68i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.836 + 0.482i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.95 - 10.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.36 + 5.81i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 11.5T + 71T^{2} \)
73 \( 1 + (-5.43 - 9.40i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.34 - 4.24i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 4.83T + 83T^{2} \)
89 \( 1 + (11.0 + 6.36i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 3.82iT - 97T^{2} \)
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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

−9.781849623186773626401145751313, −8.807868629253701813531878004301, −7.84641806524073746214992658222, −7.19473793703201941484783411232, −6.09531935667491503357386578531, −5.25572730929318743583501970965, −4.50273190126809878377588191724, −3.98477299845186948586669259576, −2.42364588946090350738108276231, −0.22090338258236417262085528738, 1.31839771022811318177316884595, 3.00443855169356524554597531334, 3.76780666509828166917759421935, 4.86770233453142220715628176872, 6.01329883015752878326138969792, 6.36964218612526623703624834506, 7.38920289559309779208133244392, 8.237071811270290066843489548712, 9.249683570569426708038791178495, 10.70500032584754270633009029128

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