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Properties

Label 2-78-39.2-c1-0-1
Degree $2$
Conductor $78$
Sign $0.973 - 0.230i$
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  + (−0.258 + 0.965i)2-s + (−0.0795 − 1.73i)3-s + (−0.866 − 0.499i)4-s + (2.02 + 2.02i)5-s + (1.69 + 0.370i)6-s + (3.46 − 0.929i)7-s + (0.707 − 0.707i)8-s + (−2.98 + 0.275i)9-s + (−2.47 + 1.42i)10-s + (−4.05 − 1.08i)11-s + (−0.796 + 1.53i)12-s + (−3.60 − 0.176i)13-s + 3.58i·14-s + (3.33 − 3.65i)15-s + (0.500 + 0.866i)16-s + (−1.72 + 2.99i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (−0.0459 − 0.998i)3-s + (−0.433 − 0.249i)4-s + (0.903 + 0.903i)5-s + (0.690 + 0.151i)6-s + (1.31 − 0.351i)7-s + (0.249 − 0.249i)8-s + (−0.995 + 0.0917i)9-s + (−0.782 + 0.451i)10-s + (−1.22 − 0.327i)11-s + (−0.229 + 0.444i)12-s + (−0.998 − 0.0490i)13-s + 0.959i·14-s + (0.861 − 0.944i)15-s + (0.125 + 0.216i)16-s + (−0.418 + 0.725i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.230i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 - 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(78\)    =    \(2 \cdot 3 \cdot 13\)
Sign: $0.973 - 0.230i$
Analytic conductor: \(0.622833\)
Root analytic conductor: \(0.789197\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{78} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 78,\ (\ :1/2),\ 0.973 - 0.230i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.924141 + 0.108079i\)
\(L(\frac12)\) \(\approx\) \(0.924141 + 0.108079i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 - 0.965i)T \)
3 \( 1 + (0.0795 + 1.73i)T \)
13 \( 1 + (3.60 + 0.176i)T \)
good5 \( 1 + (-2.02 - 2.02i)T + 5iT^{2} \)
7 \( 1 + (-3.46 + 0.929i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (4.05 + 1.08i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (1.72 - 2.99i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.581 - 2.16i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (1.51 + 2.62i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.74 + 1.00i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.21 + 1.21i)T - 31iT^{2} \)
37 \( 1 + (1.25 - 4.67i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-2.05 + 7.67i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (1.68 + 0.975i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.957 - 0.957i)T - 47iT^{2} \)
53 \( 1 - 7.22iT - 53T^{2} \)
59 \( 1 + (2.66 + 9.93i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.137 - 0.237i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.10 - 1.09i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-10.6 + 2.85i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-10.0 - 10.0i)T + 73iT^{2} \)
79 \( 1 - 1.58T + 79T^{2} \)
83 \( 1 + (2.58 + 2.58i)T + 83iT^{2} \)
89 \( 1 + (-9.50 - 2.54i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (2.07 + 7.74i)T + (-84.0 + 48.5i)T^{2} \)
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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

−14.25255225229198875297025957669, −13.83985102767818382571752684186, −12.57359975520053561972116965673, −11.04637294058986745905465341338, −10.16372580805401237619137512745, −8.328067260244481320194257290623, −7.55256144807449842199636548058, −6.35727516116318064865580285596, −5.17817326656200332643536967820, −2.24378862367465598119896816166, 2.36436601585895883167443074210, 4.83817662665721398111220542249, 5.20651396242987531879265507071, 7.977261987918138548720620181315, 9.113936995245362224996270308275, 9.913084502284390938233009411471, 11.02973295787310225494602805490, 12.05192003280170699633399518703, 13.30498566274546126356930094130, 14.34999665576343663557214245800

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