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Properties

Label 2-77-7.4-c1-0-1
Degree $2$
Conductor $77$
Sign $0.970 - 0.241i$
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  + (−0.328 − 0.568i)2-s + (−0.956 + 1.65i)3-s + (0.784 − 1.35i)4-s + (1.78 + 3.09i)5-s + 1.25·6-s + (1.78 − 1.95i)7-s − 2.34·8-s + (−0.328 − 0.568i)9-s + (1.17 − 2.02i)10-s + (0.5 − 0.866i)11-s + (1.5 + 2.59i)12-s − 5.91·13-s + (−1.69 − 0.373i)14-s − 6.82·15-s + (−0.799 − 1.38i)16-s + (0.828 − 1.43i)17-s + ⋯
L(s)  = 1  + (−0.232 − 0.402i)2-s + (−0.552 + 0.956i)3-s + (0.392 − 0.679i)4-s + (0.798 + 1.38i)5-s + 0.512·6-s + (0.674 − 0.738i)7-s − 0.828·8-s + (−0.109 − 0.189i)9-s + (0.370 − 0.641i)10-s + (0.150 − 0.261i)11-s + (0.433 + 0.749i)12-s − 1.63·13-s + (−0.453 − 0.0997i)14-s − 1.76·15-s + (−0.199 − 0.346i)16-s + (0.200 − 0.347i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $0.970 - 0.241i$
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: \((2,\ 77,\ (\ :1/2),\ 0.970 - 0.241i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.859294 + 0.105184i\)
\(L(\frac12)\) \(\approx\) \(0.859294 + 0.105184i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.78 + 1.95i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (0.328 + 0.568i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.956 - 1.65i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.78 - 3.09i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 + 5.91T + 13T^{2} \)
17 \( 1 + (-0.828 + 1.43i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.740 + 1.28i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.67 + 2.89i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.08T + 29T^{2} \)
31 \( 1 + (-3.54 + 6.13i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.25 - 3.90i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 1.28T + 41T^{2} \)
43 \( 1 - 1.59T + 43T^{2} \)
47 \( 1 + (-0.828 - 1.43i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.61 - 7.98i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.42 - 7.66i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.34 - 5.79i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.91 + 8.50i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.61T + 71T^{2} \)
73 \( 1 + (2.28 - 3.95i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.19 - 5.53i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 0.167T + 83T^{2} \)
89 \( 1 + (-1.28 - 2.22i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 9.73T + 97T^{2} \)
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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

−14.62645339215774206403624908841, −13.85622948650493308384609679217, −11.76925135691625233920694729327, −10.89049550138313444627485455202, −10.22801464080778755068184377193, −9.680411377251144313903306941087, −7.32869327540054136508016920748, −6.09766773250978124093875935884, −4.73888022017137878603403898941, −2.52994888200094150951199022014, 1.94715995869985345784970239046, 5.01040394459861023201370972214, 6.17201802612164602826761736773, 7.50413124166115584482810759714, 8.538532638271479253386403642851, 9.664929148845031817441001019589, 11.77493696254922173476409283931, 12.34134828813356808752357025542, 12.89419166521149276681781457522, 14.43881017253490563983406587454

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