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Properties

Label 1-77-77.6-r0-0-0
Degree $1$
Conductor $77$
Sign $0.0219 - 0.999i$
Analytic cond. $0.357586$
Root an. cond. $0.357586$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 − 0.587i)2-s + (−0.309 − 0.951i)3-s + (0.309 − 0.951i)4-s + (0.809 + 0.587i)5-s + (−0.809 − 0.587i)6-s + (−0.309 − 0.951i)8-s + (−0.809 + 0.587i)9-s + 10-s − 12-s + (−0.809 + 0.587i)13-s + (0.309 − 0.951i)15-s + (−0.809 − 0.587i)16-s + (−0.809 − 0.587i)17-s + (−0.309 + 0.951i)18-s + (0.309 + 0.951i)19-s + (0.809 − 0.587i)20-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)2-s + (−0.309 − 0.951i)3-s + (0.309 − 0.951i)4-s + (0.809 + 0.587i)5-s + (−0.809 − 0.587i)6-s + (−0.309 − 0.951i)8-s + (−0.809 + 0.587i)9-s + 10-s − 12-s + (−0.809 + 0.587i)13-s + (0.309 − 0.951i)15-s + (−0.809 − 0.587i)16-s + (−0.809 − 0.587i)17-s + (−0.309 + 0.951i)18-s + (0.309 + 0.951i)19-s + (0.809 − 0.587i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0219 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0219 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $0.0219 - 0.999i$
Analytic conductor: \(0.357586\)
Root analytic conductor: \(0.357586\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 77,\ (0:\ ),\ 0.0219 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9913238948 - 0.9697650696i\)
\(L(\frac12)\) \(\approx\) \(0.9913238948 - 0.9697650696i\)
\(L(1)\) \(\approx\) \(1.219732698 - 0.7668342266i\)
\(L(1)\) \(\approx\) \(1.219732698 - 0.7668342266i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.809 - 0.587i)T \)
3 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 + (0.809 + 0.587i)T \)
13 \( 1 + (-0.809 + 0.587i)T \)
17 \( 1 + (-0.809 - 0.587i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.309 + 0.951i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (0.309 - 0.951i)T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.309 - 0.951i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (-0.309 + 0.951i)T \)
61 \( 1 + (-0.809 - 0.587i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (0.809 - 0.587i)T \)
83 \( 1 + (-0.809 - 0.587i)T \)
89 \( 1 - T \)
97 \( 1 + (0.809 - 0.587i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−32.03748926478594271955658237702, −30.69168036418032878249535968486, −29.33640421689870373374637621838, −28.46188734507754438825342822638, −27.02409833093482319708181931799, −26.06643672350573742276215413713, −24.965832157281700295240287491394, −24.00021625516034222368237942798, −22.633814781454633199673892886253, −21.8617635731850582039917495496, −21.00366111877825470547864447171, −20.00006133741696880544804111423, −17.49929685517634714315747158387, −17.13170175417702845115483959496, −15.79921011650420131647259377732, −14.94295093466554678734851101208, −13.63945887374512705613801802759, −12.538670914303729435025340668448, −11.151838054458632646017193348708, −9.67439512799399461609663540920, −8.45265704371212955870829398169, −6.568733301320764713572357001242, −5.32346352798603951106336348544, −4.521477273188341623907474995450, −2.790752711064700332423271599631, 1.72612569056363330400636736493, 2.872941676570042368827334925973, 4.99191069220013488410129458335, 6.2305635075433427328844873371, 7.19006823790203154020251729714, 9.41519698387563615793830359095, 10.784753992449663968317824700728, 11.822398727821472462133223077453, 13.00232152460253927742130275969, 13.91232937686748418327031838648, 14.82050985807158446935500661687, 16.70229671702335172761754867314, 18.08568226612136036266897431267, 18.9049168119050562144389617683, 20.046160109936437452197342644033, 21.40685800606579416463694904291, 22.39278419963948652206120439105, 23.19868191333155194627159534636, 24.511257792990138944465729069040, 25.109326450604602302174206511031, 26.726297590666911171785178823565, 28.38698940291412384727711598174, 29.29170026490570206840740213516, 29.7426858177209147836021280841, 30.92695546899438077064516923007

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