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Properties

Label 2-1078-77.76-c1-0-20
Degree $2$
Conductor $1078$
Sign $0.969 + 0.245i$
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  − i·2-s + 3.12i·3-s − 4-s − 2.20i·5-s + 3.12·6-s + i·8-s − 6.77·9-s − 2.20·10-s + (−1.49 − 2.96i)11-s − 3.12i·12-s + 1.45·13-s + 6.88·15-s + 16-s + 7.60·17-s + 6.77i·18-s + 0.180·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.80i·3-s − 0.5·4-s − 0.985i·5-s + 1.27·6-s + 0.353i·8-s − 2.25·9-s − 0.696·10-s + (−0.449 − 0.893i)11-s − 0.902i·12-s + 0.404·13-s + 1.77·15-s + 0.250·16-s + 1.84·17-s + 1.59i·18-s + 0.0414·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.245i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.969 + 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $0.969 + 0.245i$
Analytic conductor: \(8.60787\)
Root analytic conductor: \(2.93391\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1078} (1077, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1078,\ (\ :1/2),\ 0.969 + 0.245i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.466554962\)
\(L(\frac12)\) \(\approx\) \(1.466554962\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
7 \( 1 \)
11 \( 1 + (1.49 + 2.96i)T \)
good3 \( 1 - 3.12iT - 3T^{2} \)
5 \( 1 + 2.20iT - 5T^{2} \)
13 \( 1 - 1.45T + 13T^{2} \)
17 \( 1 - 7.60T + 17T^{2} \)
19 \( 1 - 0.180T + 19T^{2} \)
23 \( 1 - 2.28T + 23T^{2} \)
29 \( 1 - 4.45iT - 29T^{2} \)
31 \( 1 + 8.54iT - 31T^{2} \)
37 \( 1 - 1.50T + 37T^{2} \)
41 \( 1 - 7.10T + 41T^{2} \)
43 \( 1 - 1.58iT - 43T^{2} \)
47 \( 1 + 0.545iT - 47T^{2} \)
53 \( 1 - 4.83T + 53T^{2} \)
59 \( 1 - 6.18iT - 59T^{2} \)
61 \( 1 - 5.72T + 61T^{2} \)
67 \( 1 + 2.98T + 67T^{2} \)
71 \( 1 + 7.57T + 71T^{2} \)
73 \( 1 - 9.67T + 73T^{2} \)
79 \( 1 + 6.91iT - 79T^{2} \)
83 \( 1 - 5.84T + 83T^{2} \)
89 \( 1 - 16.1iT - 89T^{2} \)
97 \( 1 + 11.3iT - 97T^{2} \)
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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

−9.855919237011697734301471044261, −9.228833047791024748046928639650, −8.594389766877162796736367667532, −7.84780587294255689352711272376, −5.78311792888843336598221779567, −5.37805277243900312732789536144, −4.47750893213666268627415137967, −3.65005137581130451810541719707, −2.87735283119832659887748699632, −0.853531050041188999740551319944, 1.09275074162423425091301645533, 2.43858004366927341766893102381, 3.41384011444780630729143898458, 5.14361084690040229689150042103, 6.01389082300084587522468712623, 6.72693980834774145108056576354, 7.41141380097874471809680900883, 7.79617657486365054848732866467, 8.710657564050139064629607061305, 9.878996739540674912484925290772

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