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Properties

Label 2-77-7.2-c1-0-4
Degree $2$
Conductor $77$
Sign $0.156 + 0.987i$
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  + (1.24 − 2.15i)2-s + (0.356 + 0.617i)3-s + (−2.10 − 3.64i)4-s + (−1.10 + 1.90i)5-s + 1.77·6-s + (−1.10 + 2.40i)7-s − 5.49·8-s + (1.24 − 2.15i)9-s + (2.74 + 4.75i)10-s + (0.5 + 0.866i)11-s + (1.5 − 2.59i)12-s − 3.28·13-s + (3.81 + 5.37i)14-s − 1.57·15-s + (−2.63 + 4.56i)16-s + (−0.745 − 1.29i)17-s + ⋯
L(s)  = 1  + (0.880 − 1.52i)2-s + (0.205 + 0.356i)3-s + (−1.05 − 1.82i)4-s + (−0.492 + 0.853i)5-s + 0.725·6-s + (−0.416 + 0.909i)7-s − 1.94·8-s + (0.415 − 0.719i)9-s + (0.868 + 1.50i)10-s + (0.150 + 0.261i)11-s + (0.433 − 0.749i)12-s − 0.911·13-s + (1.01 + 1.43i)14-s − 0.406·15-s + (−0.658 + 1.14i)16-s + (−0.180 − 0.313i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $0.156 + 0.987i$
Analytic conductor: \(0.614848\)
Root analytic conductor: \(0.784122\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 77,\ (\ :1/2),\ 0.156 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.982083 - 0.838608i\)
\(L(\frac12)\) \(\approx\) \(0.982083 - 0.838608i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (1.10 - 2.40i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (-1.24 + 2.15i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.356 - 0.617i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.10 - 1.90i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 + 3.28T + 13T^{2} \)
17 \( 1 + (0.745 + 1.29i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.45 + 5.99i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.24 - 5.62i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.64T + 29T^{2} \)
31 \( 1 + (-1.17 - 2.03i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.77 + 4.81i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 11.2T + 41T^{2} \)
43 \( 1 - 5.26T + 43T^{2} \)
47 \( 1 + (0.745 - 1.29i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.152 + 0.263i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.32 - 10.9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.49 + 11.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.28 - 3.96i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 11.3T + 71T^{2} \)
73 \( 1 + (-4.28 - 7.41i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.31 - 4.01i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.93T + 83T^{2} \)
89 \( 1 + (1.60 - 2.77i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 1.85T + 97T^{2} \)
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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

−14.13017210921567135506009457242, −12.89766498709861745785477394457, −11.94799768792793892314518957337, −11.30545903388771882752496959110, −9.923774121695763372892607427829, −9.301517888370374455056276262735, −6.99157384316780366550277920037, −5.20138593770132509944543673969, −3.70626388647952044536539493509, −2.65292920569784627899543212349, 3.96176796408239184367296603891, 5.00620061781217152127091036705, 6.55550942873089056883468716152, 7.67813894525452560083223214992, 8.311849014791559589021185830396, 10.09853856452796074907219231704, 12.16033959375269742061564579942, 12.93432408033180682324604796597, 13.80225360400661899793132600628, 14.60428036790376321168871781891

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