Properties

Label 2-1078-77.76-c1-0-0
Degree $2$
Conductor $1078$
Sign $0.381 + 0.924i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.381 + 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.381 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $0.381 + 0.924i$
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: \((2,\ 1078,\ (\ :1/2),\ 0.381 + 0.924i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.09389201200\)
\(L(\frac12)\) \(\approx\) \(0.09389201200\)
\(L(\frac{3}{2})\) 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

−10.23578067292465654793631101562, −9.600554577455351139883649847555, −8.955602404093265492117337413035, −8.416663994763515443821492731244, −7.28817528865348153289819739835, −6.06795709612365832913711192768, −5.11725283946306581158842757646, −4.57170771099171190058853440613, −4.06276665467721197613756287469, −2.50653422115915742205488504242, 0.03999687251798425386462523067, 1.56976949074715005063296826491, 2.62900331074328782307552181694, 3.19483576317159034464258206249, 4.91506777228892041352407360853, 6.07480603015160603525659126232, 6.82320221125774771537372902818, 7.34990602001853676278400265354, 8.487142622316923217392837826218, 8.895434505381250264075462926618

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