Properties

Label 2-1078-77.76-c1-0-8
Degree $2$
Conductor $1078$
Sign $0.706 - 0.707i$
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 + 1.02i·3-s − 4-s + 1.25i·5-s + 1.02·6-s + i·8-s + 1.94·9-s + 1.25·10-s + (−3.30 − 0.235i)11-s − 1.02i·12-s + 4.08·13-s − 1.29·15-s + 16-s − 3.20·17-s − 1.94i·18-s + 7.62·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.593i·3-s − 0.5·4-s + 0.562i·5-s + 0.419·6-s + 0.353i·8-s + 0.648·9-s + 0.398·10-s + (−0.997 − 0.0710i)11-s − 0.296i·12-s + 1.13·13-s − 0.333·15-s + 0.250·16-s − 0.776·17-s − 0.458i·18-s + 1.75·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.706 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.706 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $0.706 - 0.707i$
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: \((2,\ 1078,\ (\ :1/2),\ 0.706 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.464530790\)
\(L(\frac12)\) \(\approx\) \(1.464530790\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.30 + 0.235i)T \)
good3 \( 1 - 1.02iT - 3T^{2} \)
5 \( 1 - 1.25iT - 5T^{2} \)
13 \( 1 - 4.08T + 13T^{2} \)
17 \( 1 + 3.20T + 17T^{2} \)
19 \( 1 - 7.62T + 19T^{2} \)
23 \( 1 + 8.25T + 23T^{2} \)
29 \( 1 - 3.54iT - 29T^{2} \)
31 \( 1 - 9.18iT - 31T^{2} \)
37 \( 1 + 0.308T + 37T^{2} \)
41 \( 1 - 6.05T + 41T^{2} \)
43 \( 1 - 7.57iT - 43T^{2} \)
47 \( 1 - 4.70iT - 47T^{2} \)
53 \( 1 + 4.79T + 53T^{2} \)
59 \( 1 + 2.73iT - 59T^{2} \)
61 \( 1 + 1.51T + 61T^{2} \)
67 \( 1 - 3.38T + 67T^{2} \)
71 \( 1 - 3.50T + 71T^{2} \)
73 \( 1 - 0.966T + 73T^{2} \)
79 \( 1 - 15.6iT - 79T^{2} \)
83 \( 1 - 1.32T + 83T^{2} \)
89 \( 1 - 10.6iT - 89T^{2} \)
97 \( 1 + 10.6iT - 97T^{2} \)
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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

−10.06486111737702448578787092779, −9.460353918626363037445127202302, −8.481105315842264297864494417823, −7.61459893678164963772703803383, −6.60844971072376704654884117866, −5.48984717975362770059939438758, −4.61677311155788298639113143263, −3.61187301580099564916197658359, −2.84915042545349786407406957318, −1.40048319495272688002495201231, 0.72739394603641398179615028384, 2.13439456372299629726527988560, 3.74562807610733109296884376719, 4.67519751380532698821942713081, 5.68825297720326991420161684146, 6.35887063345525643484814253171, 7.50328331972117181465377693692, 7.85798267365011368660530295062, 8.772454988961750738128637013514, 9.647366459682456052684786673199

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