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Properties

Label 2-1078-77.76-c1-0-4
Degree $2$
Conductor $1078$
Sign $-0.220 - 0.975i$
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 − 0.765i·3-s − 4-s + 3.89i·5-s − 0.765·6-s + i·8-s + 2.41·9-s + 3.89·10-s + (0.214 + 3.30i)11-s + 0.765i·12-s − 1.81·13-s + 2.98·15-s + 16-s − 6.07·17-s − 2.41i·18-s − 6.82·19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.441i·3-s − 0.5·4-s + 1.74i·5-s − 0.312·6-s + 0.353i·8-s + 0.804·9-s + 1.23·10-s + (0.0647 + 0.997i)11-s + 0.220i·12-s − 0.504·13-s + 0.770·15-s + 0.250·16-s − 1.47·17-s − 0.569i·18-s − 1.56·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $-0.220 - 0.975i$
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: \((2,\ 1078,\ (\ :1/2),\ -0.220 - 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7457284383\)
\(L(\frac12)\) \(\approx\) \(0.7457284383\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.214 - 3.30i)T \)
good3 \( 1 + 0.765iT - 3T^{2} \)
5 \( 1 - 3.89iT - 5T^{2} \)
13 \( 1 + 1.81T + 13T^{2} \)
17 \( 1 + 6.07T + 17T^{2} \)
19 \( 1 + 6.82T + 19T^{2} \)
23 \( 1 + 4.37T + 23T^{2} \)
29 \( 1 + 9.36iT - 29T^{2} \)
31 \( 1 - 7.88iT - 31T^{2} \)
37 \( 1 + 4.60T + 37T^{2} \)
41 \( 1 - 3.58T + 41T^{2} \)
43 \( 1 - 3.25iT - 43T^{2} \)
47 \( 1 - 0.203iT - 47T^{2} \)
53 \( 1 - 4.98T + 53T^{2} \)
59 \( 1 - 5.19iT - 59T^{2} \)
61 \( 1 - 8.98T + 61T^{2} \)
67 \( 1 + 8.03T + 67T^{2} \)
71 \( 1 - 8.86T + 71T^{2} \)
73 \( 1 - 0.0557T + 73T^{2} \)
79 \( 1 - 8.48iT - 79T^{2} \)
83 \( 1 + 10.4T + 83T^{2} \)
89 \( 1 + 1.68iT - 89T^{2} \)
97 \( 1 - 9.23iT - 97T^{2} \)
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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

−10.23814149360758726975573142593, −9.686069548312605300156180751318, −8.458214268813957209041216529051, −7.43075544186884939667414055205, −6.82144067082635265035967334470, −6.19915314613480296990445017686, −4.53760911791551390963292991598, −3.90090503888111257043861876601, −2.41050696040170598134096126531, −2.07942385577589925890882262311, 0.31267066194336955763433577780, 1.88499082017537163409391836131, 3.93289988331120565968746248165, 4.45943330691746207204178382053, 5.24435499710239500428176449200, 6.14886076880716232446240901836, 7.14216899279154893625469644276, 8.280408365054436947015957563103, 8.744753939898263998046780080912, 9.330725004561281183489008661407

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