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Properties

Label 2-78-39.11-c1-0-0
Degree $2$
Conductor $78$
Sign $0.714 - 0.699i$
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  + (0.965 + 0.258i)2-s + (−0.933 + 1.45i)3-s + (0.866 + 0.499i)4-s + (0.313 − 0.313i)5-s + (−1.27 + 1.16i)6-s + (−0.0745 − 0.278i)7-s + (0.707 + 0.707i)8-s + (−1.25 − 2.72i)9-s + (0.383 − 0.221i)10-s + (0.150 − 0.563i)11-s + (−1.53 + 0.796i)12-s + (−1.79 − 3.12i)13-s − 0.288i·14-s + (0.164 + 0.749i)15-s + (0.500 + 0.866i)16-s + (2.79 − 4.84i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (−0.539 + 0.842i)3-s + (0.433 + 0.249i)4-s + (0.140 − 0.140i)5-s + (−0.522 + 0.476i)6-s + (−0.0281 − 0.105i)7-s + (0.249 + 0.249i)8-s + (−0.418 − 0.908i)9-s + (0.121 − 0.0700i)10-s + (0.0454 − 0.169i)11-s + (−0.444 + 0.229i)12-s + (−0.496 − 0.867i)13-s − 0.0770i·14-s + (0.0424 + 0.193i)15-s + (0.125 + 0.216i)16-s + (0.678 − 1.17i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.714 - 0.699i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.714 - 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(78\)    =    \(2 \cdot 3 \cdot 13\)
Sign: $0.714 - 0.699i$
Analytic conductor: \(0.622833\)
Root analytic conductor: \(0.789197\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{78} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 78,\ (\ :1/2),\ 0.714 - 0.699i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06522 + 0.434410i\)
\(L(\frac12)\) \(\approx\) \(1.06522 + 0.434410i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
3 \( 1 + (0.933 - 1.45i)T \)
13 \( 1 + (1.79 + 3.12i)T \)
good5 \( 1 + (-0.313 + 0.313i)T - 5iT^{2} \)
7 \( 1 + (0.0745 + 0.278i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-0.150 + 0.563i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-2.79 + 4.84i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.79 - 1.81i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-3.32 - 5.76i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.57 - 2.06i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.03 + 1.03i)T + 31iT^{2} \)
37 \( 1 + (-6.72 - 1.80i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (7.36 + 1.97i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (3.26 + 1.88i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.71 - 3.71i)T + 47iT^{2} \)
53 \( 1 - 3.64iT - 53T^{2} \)
59 \( 1 + (3.29 - 0.881i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (5.25 - 9.10i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.29 + 8.55i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (3.98 + 14.8i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-3.52 + 3.52i)T - 73iT^{2} \)
79 \( 1 - 1.10T + 79T^{2} \)
83 \( 1 + (8.23 - 8.23i)T - 83iT^{2} \)
89 \( 1 + (-3.64 + 13.6i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-2.59 + 0.694i)T + (84.0 - 48.5i)T^{2} \)
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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

−14.86997985105343226300418435161, −13.52805035776839986512806998117, −12.42636288697848300916277383074, −11.37087656375966861294482684766, −10.33189012505207288180060109770, −9.132192045436115378940864888060, −7.43106751012303519173667302840, −5.86757800305865580298043344684, −4.91133167063283160601895171717, −3.36821903224606867350158880034, 2.21148197272097829347180736101, 4.49632048024059623607826408631, 6.04554399291247481343567346538, 6.92149137627788179654641739829, 8.435873380414712935673376237797, 10.26886585423243363393464448759, 11.27904149968184824339923457473, 12.40380868970661909819600159589, 12.98483535119347807922480601862, 14.24381178992653656406020648893

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