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Properties

Label 2-1078-77.10-c1-0-35
Degree $2$
Conductor $1078$
Sign $-0.954 - 0.299i$
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 + (−2.24 − 1.29i)3-s + (0.499 − 0.866i)4-s + (3.06 − 1.76i)5-s + 2.59·6-s + 0.999i·8-s + (1.86 + 3.23i)9-s + (−1.76 + 3.06i)10-s + (−0.926 − 3.18i)11-s + (−2.24 + 1.29i)12-s − 5.01·13-s − 9.18·15-s + (−0.5 − 0.866i)16-s + (−1.94 + 3.37i)17-s + (−3.23 − 1.86i)18-s + (−2.32 − 4.03i)19-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−1.29 − 0.749i)3-s + (0.249 − 0.433i)4-s + (1.37 − 0.791i)5-s + 1.05·6-s + 0.353i·8-s + (0.622 + 1.07i)9-s + (−0.559 + 0.969i)10-s + (−0.279 − 0.960i)11-s + (−0.648 + 0.374i)12-s − 1.39·13-s − 2.37·15-s + (−0.125 − 0.216i)16-s + (−0.472 + 0.817i)17-s + (−0.762 − 0.440i)18-s + (−0.534 − 0.925i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.954 - 0.299i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.954 - 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $-0.954 - 0.299i$
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: \((2,\ 1078,\ (\ :1/2),\ -0.954 - 0.299i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2747823437\)
\(L(\frac12)\) \(\approx\) \(0.2747823437\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.926 + 3.18i)T \)
good3 \( 1 + (2.24 + 1.29i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-3.06 + 1.76i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 + 5.01T + 13T^{2} \)
17 \( 1 + (1.94 - 3.37i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.32 + 4.03i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.779 - 1.35i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 0.100iT - 29T^{2} \)
31 \( 1 + (-0.242 - 0.139i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.352 + 0.610i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.94T + 41T^{2} \)
43 \( 1 + 9.03iT - 43T^{2} \)
47 \( 1 + (5.68 - 3.28i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.77 - 4.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (12.5 + 7.26i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.56 - 9.63i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.99 - 3.45i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.45T + 71T^{2} \)
73 \( 1 + (-1.94 + 3.37i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.66 - 3.84i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 2.87T + 83T^{2} \)
89 \( 1 + (-5.15 + 2.97i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 16.7iT - 97T^{2} \)
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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

−9.346065742198140753411319138598, −8.727073888005544289735301452806, −7.63986531982194807145874256529, −6.70943074673159026103531190990, −6.06460321731664037081416734213, −5.42878633522688948957310119718, −4.73904156622939663088388436977, −2.44258993291243989754293276084, −1.39409054957845690923201087040, −0.17299783568427873261975074277, 1.90021321318706503358214563604, 2.82997493027860353445328020931, 4.49050306751148726159495540524, 5.16971046424329927084866336050, 6.17597389299535262910871716637, 6.80397089984725016322512569258, 7.72819471390700876328267870475, 9.251078276528066425402884277457, 9.863342668318864160934991153352, 10.17812189904820459711166011271

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