Properties

Label 2-77-7.4-c1-0-0
Degree $2$
Conductor $77$
Sign $-0.296 - 0.954i$
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  + (0.766 + 1.32i)2-s + (−1.43 + 2.49i)3-s + (−0.173 + 0.300i)4-s + (−1.17 − 2.03i)5-s − 4.41·6-s + (2.05 + 1.66i)7-s + 2.53·8-s + (−2.64 − 4.58i)9-s + (1.79 − 3.11i)10-s + (−0.5 + 0.866i)11-s + (−0.500 − 0.866i)12-s − 0.184·13-s + (−0.641 + 4.00i)14-s + 6.75·15-s + (2.28 + 3.96i)16-s + (1.96 − 3.39i)17-s + ⋯
L(s)  = 1  + (0.541 + 0.938i)2-s + (−0.831 + 1.43i)3-s + (−0.0868 + 0.150i)4-s + (−0.524 − 0.909i)5-s − 1.80·6-s + (0.775 + 0.630i)7-s + 0.895·8-s + (−0.881 − 1.52i)9-s + (0.568 − 0.984i)10-s + (−0.150 + 0.261i)11-s + (−0.144 − 0.250i)12-s − 0.0512·13-s + (−0.171 + 1.06i)14-s + 1.74·15-s + (0.571 + 0.990i)16-s + (0.475 − 0.823i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.296 - 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.296 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $-0.296 - 0.954i$
Analytic conductor: \(0.614848\)
Root analytic conductor: \(0.784122\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 77,\ (\ :1/2),\ -0.296 - 0.954i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.594828 + 0.807724i\)
\(L(\frac12)\) \(\approx\) \(0.594828 + 0.807724i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.05 - 1.66i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-0.766 - 1.32i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.43 - 2.49i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.17 + 2.03i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 + 0.184T + 13T^{2} \)
17 \( 1 + (-1.96 + 3.39i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.386 + 0.669i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.17 + 7.23i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 8.17T + 29T^{2} \)
31 \( 1 + (1.32 - 2.29i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.41 - 5.90i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 0.426T + 41T^{2} \)
43 \( 1 - 1.18T + 43T^{2} \)
47 \( 1 + (-3.84 - 6.65i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.27 - 5.67i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.102 - 0.177i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.29 + 12.6i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.87 - 3.25i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.96T + 71T^{2} \)
73 \( 1 + (0.0603 - 0.104i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.163 + 0.283i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.35T + 83T^{2} \)
89 \( 1 + (-2.32 - 4.02i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 13.1T + 97T^{2} \)
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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

−15.09070300023262795408899906869, −14.27772477707192179736184786673, −12.53278938521187179666173064353, −11.51426426797906827596659559994, −10.48201016882931852814588513879, −9.164621219274219571978009963916, −7.85019668045001615297489333699, −5.99534773471710112325488696400, −4.96514643585683507956602522047, −4.41215307715792759859193151860, 1.79744230101110718502144436565, 3.75229953747646392894346503755, 5.71478197185815595071563491412, 7.37264035611548788024806866697, 7.71602567121731371041422831253, 10.51477392126485567761689373149, 11.29383618462101320910625320557, 11.81859573787940580820857122867, 12.93775530499412880504680900252, 13.70597880340584772414887246157

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