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Properties

Label 2-78-39.5-c1-0-4
Degree $2$
Conductor $78$
Sign $0.999 - 0.0310i$
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 + (0.352 − 1.69i)3-s + 1.00i·4-s + (−0.499 − 0.499i)5-s + (1.44 − 0.949i)6-s + (1.39 + 1.39i)7-s + (−0.707 + 0.707i)8-s + (−2.75 − 1.19i)9-s − 0.705i·10-s + (−3.39 + 3.39i)11-s + (1.69 + 0.352i)12-s + (−2.39 − 2.69i)13-s + 1.97i·14-s + (−1.02 + 0.670i)15-s − 1.00·16-s + 4.38·17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.203 − 0.979i)3-s + 0.500i·4-s + (−0.223 − 0.223i)5-s + (0.591 − 0.387i)6-s + (0.528 + 0.528i)7-s + (−0.250 + 0.250i)8-s + (−0.916 − 0.398i)9-s − 0.223i·10-s + (−1.02 + 1.02i)11-s + (0.489 + 0.101i)12-s + (−0.665 − 0.746i)13-s + 0.528i·14-s + (−0.263 + 0.173i)15-s − 0.250·16-s + 1.06·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0310i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0310i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(78\)    =    \(2 \cdot 3 \cdot 13\)
Sign: $0.999 - 0.0310i$
Analytic conductor: \(0.622833\)
Root analytic conductor: \(0.789197\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{78} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 78,\ (\ :1/2),\ 0.999 - 0.0310i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18274 + 0.0183909i\)
\(L(\frac12)\) \(\approx\) \(1.18274 + 0.0183909i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.352 + 1.69i)T \)
13 \( 1 + (2.39 + 2.69i)T \)
good5 \( 1 + (0.499 + 0.499i)T + 5iT^{2} \)
7 \( 1 + (-1.39 - 1.39i)T + 7iT^{2} \)
11 \( 1 + (3.39 - 3.39i)T - 11iT^{2} \)
17 \( 1 - 4.38T + 17T^{2} \)
19 \( 1 + (1.70 - 1.70i)T - 19iT^{2} \)
23 \( 1 + 0.998T + 23T^{2} \)
29 \( 1 + 0.998iT - 29T^{2} \)
31 \( 1 + (-6.50 + 6.50i)T - 31iT^{2} \)
37 \( 1 + (-4.10 - 4.10i)T + 37iT^{2} \)
41 \( 1 + (-5.24 - 5.24i)T + 41iT^{2} \)
43 \( 1 - 8.88iT - 43T^{2} \)
47 \( 1 + (0.352 - 0.352i)T - 47iT^{2} \)
53 \( 1 + 14.2iT - 53T^{2} \)
59 \( 1 + (-0.998 + 0.998i)T - 59iT^{2} \)
61 \( 1 + 9.59T + 61T^{2} \)
67 \( 1 + (5.79 - 5.79i)T - 67iT^{2} \)
71 \( 1 + (7.13 + 7.13i)T + 71iT^{2} \)
73 \( 1 + (1.70 + 1.70i)T + 73iT^{2} \)
79 \( 1 + 0.207T + 79T^{2} \)
83 \( 1 + (-9.17 - 9.17i)T + 83iT^{2} \)
89 \( 1 + (-2.54 + 2.54i)T - 89iT^{2} \)
97 \( 1 + (3.58 - 3.58i)T - 97iT^{2} \)
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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

−14.63203265175365226667393628914, −13.25910100604355932545658689119, −12.49037879460743243834997966432, −11.73956778549319446338487541750, −9.928290778793127880461919992862, −8.070512401414089130228438668706, −7.77166933207939510687873159385, −6.12280122949724851640943093104, −4.85920092912969207655615249622, −2.58061358722273534316177629584, 2.98354879085718255505348241214, 4.39666394941560438579021476071, 5.59306071394779687634853653645, 7.62242054668809914822230571460, 9.017779496211895769090585892608, 10.39245232860030462040959725117, 10.96414730470624087704116206867, 12.11403955135732743447431686375, 13.69544548474860964886954928868, 14.30142785525974245584870728705

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