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Properties

Label 2-77-11.5-c1-0-4
Degree $2$
Conductor $77$
Sign $0.565 + 0.824i$
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  + (0.206 + 0.636i)2-s + (−2.54 − 1.84i)3-s + (1.25 − 0.911i)4-s + (0.662 − 2.03i)5-s + (0.649 − 1.99i)6-s + (−0.809 + 0.587i)7-s + (1.92 + 1.39i)8-s + (2.11 + 6.52i)9-s + 1.43·10-s + (−1.08 − 3.13i)11-s − 4.87·12-s + (0.781 + 2.40i)13-s + (−0.541 − 0.393i)14-s + (−5.44 + 3.95i)15-s + (0.466 − 1.43i)16-s + (−0.553 + 1.70i)17-s + ⋯
L(s)  = 1  + (0.146 + 0.450i)2-s + (−1.46 − 1.06i)3-s + (0.627 − 0.455i)4-s + (0.296 − 0.911i)5-s + (0.265 − 0.816i)6-s + (−0.305 + 0.222i)7-s + (0.680 + 0.494i)8-s + (0.706 + 2.17i)9-s + 0.454·10-s + (−0.326 − 0.945i)11-s − 1.40·12-s + (0.216 + 0.666i)13-s + (−0.144 − 0.105i)14-s + (−1.40 + 1.02i)15-s + (0.116 − 0.358i)16-s + (−0.134 + 0.413i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.565 + 0.824i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.565 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $0.565 + 0.824i$
Analytic conductor: \(0.614848\)
Root analytic conductor: \(0.784122\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 77,\ (\ :1/2),\ 0.565 + 0.824i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.704409 - 0.370896i\)
\(L(\frac12)\) \(\approx\) \(0.704409 - 0.370896i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 + (1.08 + 3.13i)T \)
good2 \( 1 + (-0.206 - 0.636i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (2.54 + 1.84i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (-0.662 + 2.03i)T + (-4.04 - 2.93i)T^{2} \)
13 \( 1 + (-0.781 - 2.40i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.553 - 1.70i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-5.44 - 3.95i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 3.16T + 23T^{2} \)
29 \( 1 + (-0.747 + 0.543i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.927 - 2.85i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-1.21 + 0.885i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (4.49 + 3.26i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 8.42T + 43T^{2} \)
47 \( 1 + (3.55 + 2.58i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.206 - 0.634i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-0.298 + 0.216i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-1.54 + 4.76i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 0.902T + 67T^{2} \)
71 \( 1 + (4.59 - 14.1i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (6.50 - 4.72i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-1.25 - 3.85i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (1.25 - 3.85i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 8.30T + 89T^{2} \)
97 \( 1 + (-2.63 - 8.09i)T + (-78.4 + 57.0i)T^{2} \)
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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

−14.04480193235083829651657821970, −13.16725454647950063016754770945, −12.06502017499900817292225897526, −11.35974591959427147182465013853, −10.14910643581265185260181800545, −8.231561609695497562241012163505, −6.88688450083252479181897657755, −5.89811242223099726150500641130, −5.23268932195000846494486824193, −1.46380038616775393221737670558, 3.18143608442819947579568218588, 4.76547158668519222644871402436, 6.27779942561270515532272026516, 7.27645652241244805496874811753, 9.833866361394398864386088270901, 10.37713993296008570891735021669, 11.31320647160512938328659839905, 12.04800322577296559973969427967, 13.29212391176487730449952046549, 15.05997992501589149578903411153

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