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Properties

Label 2-777-777.101-c1-0-48
Degree $2$
Conductor $777$
Sign $0.697 + 0.716i$
Analytic cond. $6.20437$
Root an. cond. $2.49085$
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  + 1.73i·3-s − 2·4-s + (−2 − 1.73i)7-s − 2.99·9-s − 3.46i·12-s + (1 + 1.73i)13-s + 4·16-s + (3.5 − 6.06i)19-s + (2.99 − 3.46i)21-s + (−2.5 − 4.33i)25-s − 5.19i·27-s + (4 + 3.46i)28-s + (2 − 3.46i)31-s + 5.99·36-s + (5 − 3.46i)37-s + ⋯
L(s)  = 1  + 0.999i·3-s − 4-s + (−0.755 − 0.654i)7-s − 0.999·9-s − 0.999i·12-s + (0.277 + 0.480i)13-s + 16-s + (0.802 − 1.39i)19-s + (0.654 − 0.755i)21-s + (−0.5 − 0.866i)25-s − 0.999i·27-s + (0.755 + 0.654i)28-s + (0.359 − 0.622i)31-s + 0.999·36-s + (0.821 − 0.569i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 777 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.697 + 0.716i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.697 + 0.716i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(777\)    =    \(3 \cdot 7 \cdot 37\)
Sign: $0.697 + 0.716i$
Analytic conductor: \(6.20437\)
Root analytic conductor: \(2.49085\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{777} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 777,\ (\ :1/2),\ 0.697 + 0.716i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.700707 - 0.295794i\)
\(L(\frac12)\) \(\approx\) \(0.700707 - 0.295794i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 1.73iT \)
7 \( 1 + (2 + 1.73i)T \)
37 \( 1 + (-5 + 3.46i)T \)
good2 \( 1 + 2T^{2} \)
5 \( 1 + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1 - 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.5 + 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 1.73iT - 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7 + 12.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 - 16T + 67T^{2} \)
71 \( 1 + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (12 + 6.92i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (15 + 8.66i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + 19T + 97T^{2} \)
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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

−9.885263366114113452008666727557, −9.549742246656140919643155742855, −8.763061513352824345524696761918, −7.81012622966164620292617883531, −6.58599723602549628109480463623, −5.56756781800964845637403600258, −4.54569734401035336000842802753, −3.94316772696914432880306835692, −2.90010432838798019437564602629, −0.45312227027854184447384262704, 1.22943307150464249275969380415, 2.86173165378875759116575484023, 3.80903686958232423216235871790, 5.42771141803457542641790543433, 5.82170899407284218869629900321, 6.97540197310310198222060402745, 8.007982848200112035410719532157, 8.571281317405663989893582541517, 9.512243829071499683183242705188, 10.17543921108565168963988433819

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