Properties

Label 2-78-39.8-c1-0-5
Degree $2$
Conductor $78$
Sign $0.682 + 0.731i$
Analytic cond. $0.622833$
Root an. cond. $0.789197$
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  + (0.707 − 0.707i)2-s + (1.29 − 1.15i)3-s − 1.00i·4-s + (−1.82 + 1.82i)5-s + (0.0980 − 1.72i)6-s + (−2.63 + 2.63i)7-s + (−0.707 − 0.707i)8-s + (0.339 − 2.98i)9-s + 2.58i·10-s + (2.30 + 2.30i)11-s + (−1.15 − 1.29i)12-s + (1.63 − 3.21i)13-s + 3.72i·14-s + (−0.253 + 4.46i)15-s − 1.00·16-s + 1.34·17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (0.746 − 0.665i)3-s − 0.500i·4-s + (−0.817 + 0.817i)5-s + (0.0400 − 0.705i)6-s + (−0.994 + 0.994i)7-s + (−0.250 − 0.250i)8-s + (0.113 − 0.993i)9-s + 0.817i·10-s + (0.695 + 0.695i)11-s + (−0.332 − 0.373i)12-s + (0.452 − 0.891i)13-s + 0.994i·14-s + (−0.0654 + 1.15i)15-s − 0.250·16-s + 0.326·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.682 + 0.731i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.682 + 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(78\)    =    \(2 \cdot 3 \cdot 13\)
Sign: $0.682 + 0.731i$
Analytic conductor: \(0.622833\)
Root analytic conductor: \(0.789197\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{78} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 78,\ (\ :1/2),\ 0.682 + 0.731i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13319 - 0.492603i\)
\(L(\frac12)\) \(\approx\) \(1.13319 - 0.492603i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (-1.29 + 1.15i)T \)
13 \( 1 + (-1.63 + 3.21i)T \)
good5 \( 1 + (1.82 - 1.82i)T - 5iT^{2} \)
7 \( 1 + (2.63 - 2.63i)T - 7iT^{2} \)
11 \( 1 + (-2.30 - 2.30i)T + 11iT^{2} \)
17 \( 1 - 1.34T + 17T^{2} \)
19 \( 1 + (3.58 + 3.58i)T + 19iT^{2} \)
23 \( 1 + 3.65T + 23T^{2} \)
29 \( 1 - 3.65iT - 29T^{2} \)
31 \( 1 + (-0.321 - 0.321i)T + 31iT^{2} \)
37 \( 1 + (-1.95 + 1.95i)T - 37iT^{2} \)
41 \( 1 + (-7.89 + 7.89i)T - 41iT^{2} \)
43 \( 1 - 9.10iT - 43T^{2} \)
47 \( 1 + (4.72 + 4.72i)T + 47iT^{2} \)
53 \( 1 - 0.216iT - 53T^{2} \)
59 \( 1 + (-3.65 - 3.65i)T + 59iT^{2} \)
61 \( 1 - 6.52T + 61T^{2} \)
67 \( 1 + (-2.26 - 2.26i)T + 67iT^{2} \)
71 \( 1 + (0.108 - 0.108i)T - 71iT^{2} \)
73 \( 1 + (3.58 - 3.58i)T - 73iT^{2} \)
79 \( 1 - 4.09T + 79T^{2} \)
83 \( 1 + (10.5 - 10.5i)T - 83iT^{2} \)
89 \( 1 + (8.85 + 8.85i)T + 89iT^{2} \)
97 \( 1 + (-0.168 - 0.168i)T + 97iT^{2} \)
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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

−14.39388274271234461853671547757, −12.96295015053351104317312354435, −12.41209466469793658678213060442, −11.32607936889717867582347247068, −9.857261152686133987627714830765, −8.681039257925309446258539375383, −7.19490331665694771052597013474, −6.12321585070243764219511816914, −3.74591266769627696479587624501, −2.63780849197529667581591007760, 3.70342917805571943037463869945, 4.28252569387665086331582308519, 6.30425355357628629967355252549, 7.82053394099028792728402879788, 8.766767441583590823876784739769, 9.959438482459977538170504104459, 11.44370256022451438679544208599, 12.74234387577194075402683591769, 13.72304884558133805580121193719, 14.49926741198064689873331433255

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