Properties

Label 2-1078-77.76-c1-0-5
Degree $2$
Conductor $1078$
Sign $-0.0929 + 0.995i$
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  + i·2-s + 2.59i·3-s − 4-s + 3.53i·5-s − 2.59·6-s i·8-s − 3.73·9-s − 3.53·10-s + (−2.29 − 2.39i)11-s − 2.59i·12-s + 5.01·13-s − 9.18·15-s + 16-s − 3.89·17-s − 3.73i·18-s − 4.65·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.49i·3-s − 0.5·4-s + 1.58i·5-s − 1.05·6-s − 0.353i·8-s − 1.24·9-s − 1.11·10-s + (−0.691 − 0.722i)11-s − 0.749i·12-s + 1.39·13-s − 2.37·15-s + 0.250·16-s − 0.944·17-s − 0.880i·18-s − 1.06·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0929 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0929 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $-0.0929 + 0.995i$
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: \((2,\ 1078,\ (\ :1/2),\ -0.0929 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9505748666\)
\(L(\frac12)\) \(\approx\) \(0.9505748666\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.29 + 2.39i)T \)
good3 \( 1 - 2.59iT - 3T^{2} \)
5 \( 1 - 3.53iT - 5T^{2} \)
13 \( 1 - 5.01T + 13T^{2} \)
17 \( 1 + 3.89T + 17T^{2} \)
19 \( 1 + 4.65T + 19T^{2} \)
23 \( 1 + 1.55T + 23T^{2} \)
29 \( 1 - 0.100iT - 29T^{2} \)
31 \( 1 + 0.279iT - 31T^{2} \)
37 \( 1 - 0.705T + 37T^{2} \)
41 \( 1 + 2.94T + 41T^{2} \)
43 \( 1 - 9.03iT - 43T^{2} \)
47 \( 1 + 6.56iT - 47T^{2} \)
53 \( 1 - 5.54T + 53T^{2} \)
59 \( 1 - 14.5iT - 59T^{2} \)
61 \( 1 - 11.1T + 61T^{2} \)
67 \( 1 - 3.98T + 67T^{2} \)
71 \( 1 + 8.45T + 71T^{2} \)
73 \( 1 - 3.89T + 73T^{2} \)
79 \( 1 - 7.69iT - 79T^{2} \)
83 \( 1 + 2.87T + 83T^{2} \)
89 \( 1 - 5.94iT - 89T^{2} \)
97 \( 1 - 16.7iT - 97T^{2} \)
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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

−10.59739504690659069369552966405, −9.774194291735241392970101756982, −8.748508854553918545949352335969, −8.212367647782665426369671571474, −6.94746381891821354289195869070, −6.23319758649646776497272919589, −5.49795497516921102363306002781, −4.24985005262363860645089821518, −3.62983317916439579912238055163, −2.64669639839000638378168447601, 0.42182088750560705204079943112, 1.56717421279204087130398551983, 2.25482807113572142830130905864, 3.94035179205739497758979327305, 4.85822194331716285513305229977, 5.85063719646376929737277687267, 6.75648674646621887295117196758, 7.88054858522714119664394900547, 8.510078998966810547865965120495, 8.968758418907028410877901785101

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