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Properties

Label 2-78-39.20-c1-0-0
Degree $2$
Conductor $78$
Sign $-0.373 - 0.927i$
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 + (−1.45 + 0.933i)3-s + (−0.866 + 0.499i)4-s + (−2.02 + 2.02i)5-s + (−1.27 − 1.16i)6-s + (3.46 + 0.929i)7-s + (−0.707 − 0.707i)8-s + (1.25 − 2.72i)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 + (1.06 − 4.83i)15-s + (0.500 − 0.866i)16-s + (1.72 + 2.99i)17-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (−0.842 + 0.539i)3-s + (−0.433 + 0.249i)4-s + (−0.903 + 0.903i)5-s + (−0.522 − 0.476i)6-s + (1.31 + 0.351i)7-s + (−0.249 − 0.249i)8-s + (0.418 − 0.908i)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 + (0.273 − 1.24i)15-s + (0.125 − 0.216i)16-s + (0.418 + 0.725i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.373 - 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.373 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(78\)    =    \(2 \cdot 3 \cdot 13\)
Sign: $-0.373 - 0.927i$
Analytic conductor: \(0.622833\)
Root analytic conductor: \(0.789197\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{78} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 78,\ (\ :1/2),\ -0.373 - 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.433217 + 0.641344i\)
\(L(\frac12)\) \(\approx\) \(0.433217 + 0.641344i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.45 - 0.933i)T \)
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

−14.81991138872175130034663368917, −14.36683371845398085121202802019, −12.26668620455312118614150866995, −11.56510017879757605075435837467, −10.65229074303146141476932284530, −9.051330154245990281004535961796, −7.65262014659727648802297335759, −6.54990046093244308338979486406, −5.09230385833849599776583081698, −3.86856995791394296230245012185, 1.31994732191858571724124188623, 4.33861600746475468326518792472, 5.17896898050742297987606934607, 7.23319642229864062345135020098, 8.293483157278294185601681381058, 9.868658593760215567117000642475, 11.46646267030499352843651622909, 11.75138188651280551331487761974, 12.62466625153211681642075977432, 13.95127112514186933297377284137

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