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Properties

Label 2-77-77.10-c1-0-4
Degree $2$
Conductor $77$
Sign $0.443 + 0.896i$
Analytic cond. $0.614848$
Root an. cond. $0.784122$
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  + (1.43 − 0.830i)2-s + (−1.97 − 1.13i)3-s + (0.380 − 0.658i)4-s + (2.80 − 1.61i)5-s − 3.78·6-s + (−2.23 + 1.41i)7-s + 2.05i·8-s + (1.09 + 1.88i)9-s + (2.68 − 4.65i)10-s + (2.41 + 2.27i)11-s + (−1.50 + 0.866i)12-s − 0.904·13-s + (−2.04 + 3.89i)14-s − 7.36·15-s + (2.47 + 4.28i)16-s + (−1.78 + 3.08i)17-s + ⋯
L(s)  = 1  + (1.01 − 0.587i)2-s + (−1.13 − 0.657i)3-s + (0.190 − 0.329i)4-s + (1.25 − 0.723i)5-s − 1.54·6-s + (−0.844 + 0.534i)7-s + 0.727i·8-s + (0.363 + 0.629i)9-s + (0.849 − 1.47i)10-s + (0.728 + 0.685i)11-s + (−0.433 + 0.249i)12-s − 0.250·13-s + (−0.545 + 1.04i)14-s − 1.90·15-s + (0.617 + 1.07i)16-s + (−0.432 + 0.749i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $0.443 + 0.896i$
Analytic conductor: \(0.614848\)
Root analytic conductor: \(0.784122\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 77,\ (\ :1/2),\ 0.443 + 0.896i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00147 - 0.621641i\)
\(L(\frac12)\) \(\approx\) \(1.00147 - 0.621641i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.23 - 1.41i)T \)
11 \( 1 + (-2.41 - 2.27i)T \)
good2 \( 1 + (-1.43 + 0.830i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (1.97 + 1.13i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-2.80 + 1.61i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 + 0.904T + 13T^{2} \)
17 \( 1 + (1.78 - 3.08i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.99 + 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.09 + 5.35i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 7.34iT - 29T^{2} \)
31 \( 1 + (-0.358 - 0.207i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.23 - 2.14i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.71T + 41T^{2} \)
43 \( 1 + 2.15iT - 43T^{2} \)
47 \( 1 + (3.11 - 1.79i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.39 - 7.60i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-9.29 - 5.36i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.97 - 3.41i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.660 + 1.14i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.70T + 71T^{2} \)
73 \( 1 + (-3.67 + 6.36i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.67 + 3.27i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 15.0T + 83T^{2} \)
89 \( 1 + (2.41 - 1.39i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 6.82iT - 97T^{2} \)
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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

−13.75507818665799261944553087814, −12.83763370175591489446231089317, −12.48512180953955662204249560672, −11.53173487485246416730090924513, −10.08289351429691507189597483454, −8.815166323084592398921954895913, −6.49334495055318489016392612873, −5.81395739130730737175196096529, −4.54756982865737202031455957636, −2.14999580732140385839656985964, 3.66251940651857310654816923249, 5.24674185277199690771410279421, 6.14906495226792931741314961844, 6.81000808823320491544923714254, 9.600435135578276161658195525199, 10.20843825770124545742918599458, 11.37957186516267016790603362516, 12.81402239272804120597992837701, 13.85209768175137211977892722389, 14.42244523570154894746588989296

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