Properties

Label 2-77-77.19-c1-0-5
Degree $2$
Conductor $77$
Sign $0.713 + 0.700i$
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  + (2.05 − 0.216i)2-s + (−2.33 − 2.10i)3-s + (2.23 − 0.475i)4-s + (0.483 + 1.08i)5-s + (−5.26 − 3.82i)6-s + (2.02 + 1.70i)7-s + (0.563 − 0.183i)8-s + (0.717 + 6.83i)9-s + (1.22 + 2.13i)10-s + (−2.42 − 2.26i)11-s + (−6.22 − 3.59i)12-s + (0.000347 − 0.000252i)13-s + (4.53 + 3.07i)14-s + (1.15 − 3.54i)15-s + (−3.05 + 1.36i)16-s + (−0.192 + 1.83i)17-s + ⋯
L(s)  = 1  + (1.45 − 0.153i)2-s + (−1.34 − 1.21i)3-s + (1.11 − 0.237i)4-s + (0.216 + 0.485i)5-s + (−2.14 − 1.56i)6-s + (0.764 + 0.645i)7-s + (0.199 − 0.0647i)8-s + (0.239 + 2.27i)9-s + (0.388 + 0.673i)10-s + (−0.731 − 0.682i)11-s + (−1.79 − 1.03i)12-s + (9.63e−5 − 6.99e−5i)13-s + (1.21 + 0.822i)14-s + (0.297 − 0.916i)15-s + (−0.763 + 0.340i)16-s + (−0.0467 + 0.444i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.713 + 0.700i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.713 + 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $0.713 + 0.700i$
Analytic conductor: \(0.614848\)
Root analytic conductor: \(0.784122\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 77,\ (\ :1/2),\ 0.713 + 0.700i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22574 - 0.501011i\)
\(L(\frac12)\) \(\approx\) \(1.22574 - 0.501011i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.02 - 1.70i)T \)
11 \( 1 + (2.42 + 2.26i)T \)
good2 \( 1 + (-2.05 + 0.216i)T + (1.95 - 0.415i)T^{2} \)
3 \( 1 + (2.33 + 2.10i)T + (0.313 + 2.98i)T^{2} \)
5 \( 1 + (-0.483 - 1.08i)T + (-3.34 + 3.71i)T^{2} \)
13 \( 1 + (-0.000347 + 0.000252i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.192 - 1.83i)T + (-16.6 - 3.53i)T^{2} \)
19 \( 1 + (1.89 + 0.402i)T + (17.3 + 7.72i)T^{2} \)
23 \( 1 + (-2.83 + 4.91i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.88 + 1.58i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-0.176 + 0.396i)T + (-20.7 - 23.0i)T^{2} \)
37 \( 1 + (-0.251 - 0.279i)T + (-3.86 + 36.7i)T^{2} \)
41 \( 1 + (-0.760 - 2.34i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 5.03iT - 43T^{2} \)
47 \( 1 + (0.385 - 1.81i)T + (-42.9 - 19.1i)T^{2} \)
53 \( 1 + (-1.20 - 0.538i)T + (35.4 + 39.3i)T^{2} \)
59 \( 1 + (2.53 + 11.9i)T + (-53.8 + 23.9i)T^{2} \)
61 \( 1 + (-6.95 + 3.09i)T + (40.8 - 45.3i)T^{2} \)
67 \( 1 + (-6.72 - 11.6i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.82 + 4.96i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-13.3 + 2.83i)T + (66.6 - 29.6i)T^{2} \)
79 \( 1 + (-1.63 + 0.171i)T + (77.2 - 16.4i)T^{2} \)
83 \( 1 + (-7.86 - 5.71i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (6.24 + 3.60i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.39 + 3.30i)T + (-29.9 + 92.2i)T^{2} \)
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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

−14.06450330657304009553938278500, −13.00319486289975568732235080657, −12.46287652716394236854748807046, −11.32069376704540264357716051267, −10.86351424652025496580195628090, −8.198009135285957854632622358022, −6.64483724932407147863183967569, −5.82492032691226422511245937296, −4.88241785352517956705569991488, −2.39204917892641940246386736987, 3.92227182700382075855350603967, 4.98427512170840317854928136121, 5.45965051780739170945697079942, 7.06087027711061070269136551832, 9.346183531668247136360008179889, 10.62016620228318199702691380334, 11.45783343287259108488203778577, 12.46758472932772000607356857645, 13.48296319007258404198826298530, 14.84388103038786122854540285322

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