Properties

Label 2-1078-77.76-c1-0-3
Degree $2$
Conductor $1078$
Sign $-0.562 - 0.826i$
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 + 2.61i·3-s − 4-s − 2.61i·5-s + 2.61·6-s + i·8-s − 3.82·9-s − 2.61·10-s + (1.41 + 3i)11-s − 2.61i·12-s − 5.54·13-s + 6.82·15-s + 16-s + 3.82i·18-s − 2.29·19-s + 2.61i·20-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.50i·3-s − 0.5·4-s − 1.16i·5-s + 1.06·6-s + 0.353i·8-s − 1.27·9-s − 0.826·10-s + (0.426 + 0.904i)11-s − 0.754i·12-s − 1.53·13-s + 1.76·15-s + 0.250·16-s + 0.902i·18-s − 0.526·19-s + 0.584i·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.562 - 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.562 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $-0.562 - 0.826i$
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: \((2,\ 1078,\ (\ :1/2),\ -0.562 - 0.826i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6866117141\)
\(L(\frac12)\) \(\approx\) \(0.6866117141\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.41 - 3i)T \)
good3 \( 1 - 2.61iT - 3T^{2} \)
5 \( 1 + 2.61iT - 5T^{2} \)
13 \( 1 + 5.54T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 2.29T + 19T^{2} \)
23 \( 1 + 2.24T + 23T^{2} \)
29 \( 1 - 10.2iT - 29T^{2} \)
31 \( 1 - 7.07iT - 31T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 + 11.0T + 41T^{2} \)
43 \( 1 + 4.24iT - 43T^{2} \)
47 \( 1 - 6.62iT - 47T^{2} \)
53 \( 1 + 12.4T + 53T^{2} \)
59 \( 1 + 12.1iT - 59T^{2} \)
61 \( 1 + 2.29T + 61T^{2} \)
67 \( 1 - 0.343T + 67T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + 6.49T + 73T^{2} \)
79 \( 1 - 8.48iT - 79T^{2} \)
83 \( 1 - 0.951T + 83T^{2} \)
89 \( 1 - 12.3iT - 89T^{2} \)
97 \( 1 + 5.09iT - 97T^{2} \)
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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

−10.09783301332224894388521381631, −9.408524716883970127370532366634, −9.006465011152017465104755559095, −8.066204050838795834111403162205, −6.79801443612339120887983178314, −5.10375210211732710243520898418, −4.95027399959015322639988802610, −4.17402483941321831581903643164, −3.11029106768976452479271521201, −1.68587529569624291891029104163, 0.29429501799943452744540486802, 2.08814824533555004383526653048, 3.00655166299228507508366607090, 4.40588201217524698434149434653, 5.89823712457578864445179315484, 6.32682382832000677485711680144, 7.09581136443754980738629573899, 7.71357956071731183825282770869, 8.318178414438514577018026518827, 9.533451367146359238837902499003

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