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Properties

Label 2-1078-77.76-c1-0-17
Degree $2$
Conductor $1078$
Sign $-0.638 - 0.769i$
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 + 2.05i·5-s − 0.765·6-s − i·8-s + 2.41·9-s − 2.05·10-s + (2.49 + 2.18i)11-s − 0.765i·12-s + 6.59·13-s − 1.57·15-s + 16-s − 3.61·17-s + 2.41i·18-s − 0.878·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.441i·3-s − 0.5·4-s + 0.917i·5-s − 0.312·6-s − 0.353i·8-s + 0.804·9-s − 0.648·10-s + (0.751 + 0.659i)11-s − 0.220i·12-s + 1.82·13-s − 0.405·15-s + 0.250·16-s − 0.875·17-s + 0.569i·18-s − 0.201·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.638 - 0.769i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.638 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1078\)    =    \(2 \cdot 7^{2} \cdot 11\)
Sign: $-0.638 - 0.769i$
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: \((2,\ 1078,\ (\ :1/2),\ -0.638 - 0.769i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.791215083\)
\(L(\frac12)\) \(\approx\) \(1.791215083\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.49 - 2.18i)T \)
good3 \( 1 - 0.765iT - 3T^{2} \)
5 \( 1 - 2.05iT - 5T^{2} \)
13 \( 1 - 6.59T + 13T^{2} \)
17 \( 1 + 3.61T + 17T^{2} \)
19 \( 1 + 0.878T + 19T^{2} \)
23 \( 1 - 6.61T + 23T^{2} \)
29 \( 1 + 6.18iT - 29T^{2} \)
31 \( 1 - 6.48iT - 31T^{2} \)
37 \( 1 + 11.0T + 37T^{2} \)
41 \( 1 + 2.36T + 41T^{2} \)
43 \( 1 + 7.81iT - 43T^{2} \)
47 \( 1 - 5.74iT - 47T^{2} \)
53 \( 1 - 0.429T + 53T^{2} \)
59 \( 1 - 3.21iT - 59T^{2} \)
61 \( 1 + 11.3T + 61T^{2} \)
67 \( 1 - 2.96T + 67T^{2} \)
71 \( 1 + 2.13T + 71T^{2} \)
73 \( 1 + 10.8T + 73T^{2} \)
79 \( 1 + 8.48iT - 79T^{2} \)
83 \( 1 - 12.3T + 83T^{2} \)
89 \( 1 + 6.72iT - 89T^{2} \)
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

−10.33895521124298335436544875957, −9.049345887044789334923625582734, −8.788187675058389709947813365898, −7.41300958127017388702670086180, −6.77454876732274544589532272923, −6.25374917530027681542314962980, −4.95533604401538626977984794748, −4.07849012222925590495605930516, −3.27794749074032375796729623230, −1.53783445611018597141774004930, 0.970006485809199657894570528139, 1.65499261891564630094230825829, 3.31761905737208812877840171082, 4.16626590178864682710087430265, 5.08730900298785380611818256908, 6.23368708802856329444609941618, 6.98538646402126532674153912548, 8.323761295521897803769436949985, 8.803428736626751846582741005234, 9.385938411639874499782352733387

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