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Properties

Label 2-1078-77.10-c1-0-4
Degree $2$
Conductor $1078$
Sign $-0.0910 - 0.995i$
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 + (−1.35 − 0.779i)3-s + (0.499 − 0.866i)4-s + (−0.882 + 0.509i)5-s − 1.55·6-s − 0.999i·8-s + (−0.284 − 0.492i)9-s + (−0.509 + 0.882i)10-s + (−2.58 + 2.08i)11-s + (−1.35 + 0.779i)12-s − 0.167·13-s + 1.58·15-s + (−0.5 − 0.866i)16-s + (−1.47 + 2.55i)17-s + (−0.492 − 0.284i)18-s + (−0.155 − 0.269i)19-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.779 − 0.450i)3-s + (0.249 − 0.433i)4-s + (−0.394 + 0.227i)5-s − 0.636·6-s − 0.353i·8-s + (−0.0947 − 0.164i)9-s + (−0.161 + 0.279i)10-s + (−0.778 + 0.627i)11-s + (−0.389 + 0.225i)12-s − 0.0463·13-s + 0.410·15-s + (−0.125 − 0.216i)16-s + (−0.358 + 0.620i)17-s + (−0.116 − 0.0670i)18-s + (−0.0357 − 0.0619i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0910 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0910 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $-0.0910 - 0.995i$
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: \((2,\ 1078,\ (\ :1/2),\ -0.0910 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4338814588\)
\(L(\frac12)\) \(\approx\) \(0.4338814588\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.58 - 2.08i)T \)
good3 \( 1 + (1.35 + 0.779i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.882 - 0.509i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 + 0.167T + 13T^{2} \)
17 \( 1 + (1.47 - 2.55i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.155 + 0.269i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.237 - 0.411i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.89iT - 29T^{2} \)
31 \( 1 + (-2.20 - 1.27i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.04 + 5.27i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 10.2T + 41T^{2} \)
43 \( 1 - 10.1iT - 43T^{2} \)
47 \( 1 + (-3.28 + 1.89i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.21 - 7.29i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.40 + 3.12i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.93 - 10.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.19 + 8.99i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 14.5T + 71T^{2} \)
73 \( 1 + (4.85 - 8.40i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.06 - 3.50i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 14.6T + 83T^{2} \)
89 \( 1 + (5.97 - 3.45i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 11.0iT - 97T^{2} \)
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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

−10.38734108599370766158984844661, −9.471208414710130075944067929727, −8.345146325446991458347773553825, −7.31531744285157343418354251564, −6.67891448696588760077841031590, −5.74832494337798475292249054251, −5.01091927756277459226444088953, −3.94332885489629942009394777069, −2.87950316092447523025017121107, −1.54815776409799270004565621062, 0.16811917301608920703580924438, 2.42003462193485422101926746770, 3.61635180947655728706912320060, 4.69307323757684689697802621811, 5.22885732274317225363376326875, 6.07092645528431050943992409229, 6.97565864029934316327796886656, 8.056018418343615229365579079438, 8.547409199397271948907679554956, 9.893929359680316579664988247339

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