Properties

Label 2-77-77.13-c1-0-4
Degree $2$
Conductor $77$
Sign $-0.942 + 0.335i$
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  + (−1.16 − 0.379i)2-s + (−0.767 − 1.05i)3-s + (−0.395 − 0.287i)4-s + (−2.26 + 0.737i)5-s + (0.496 + 1.52i)6-s + (−2.39 − 1.12i)7-s + (1.79 + 2.47i)8-s + (0.400 − 1.23i)9-s + 2.93·10-s + (−2.38 − 2.30i)11-s + 0.638i·12-s + (0.802 − 2.47i)13-s + (2.37 + 2.22i)14-s + (2.52 + 1.83i)15-s + (−0.860 − 2.64i)16-s + (1.40 + 4.31i)17-s + ⋯
L(s)  = 1  + (−0.826 − 0.268i)2-s + (−0.443 − 0.609i)3-s + (−0.197 − 0.143i)4-s + (−1.01 + 0.329i)5-s + (0.202 + 0.623i)6-s + (−0.905 − 0.423i)7-s + (0.635 + 0.875i)8-s + (0.133 − 0.410i)9-s + 0.927·10-s + (−0.717 − 0.696i)11-s + 0.184i·12-s + (0.222 − 0.685i)13-s + (0.635 + 0.593i)14-s + (0.650 + 0.472i)15-s + (−0.215 − 0.661i)16-s + (0.340 + 1.04i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.942 + 0.335i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.942 + 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $-0.942 + 0.335i$
Analytic conductor: \(0.614848\)
Root analytic conductor: \(0.784122\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 77,\ (\ :1/2),\ -0.942 + 0.335i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0453055 - 0.262619i\)
\(L(\frac12)\) \(\approx\) \(0.0453055 - 0.262619i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.39 + 1.12i)T \)
11 \( 1 + (2.38 + 2.30i)T \)
good2 \( 1 + (1.16 + 0.379i)T + (1.61 + 1.17i)T^{2} \)
3 \( 1 + (0.767 + 1.05i)T + (-0.927 + 2.85i)T^{2} \)
5 \( 1 + (2.26 - 0.737i)T + (4.04 - 2.93i)T^{2} \)
13 \( 1 + (-0.802 + 2.47i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.40 - 4.31i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-6.65 + 4.83i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 3.85T + 23T^{2} \)
29 \( 1 + (-1.65 + 2.28i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (3.93 + 1.27i)T + (25.0 + 18.2i)T^{2} \)
37 \( 1 + (3.25 + 2.36i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (0.654 - 0.475i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 1.73iT - 43T^{2} \)
47 \( 1 + (1.08 + 1.48i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (0.773 - 2.38i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-6.33 + 8.71i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (-1.91 - 5.90i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 11.5T + 67T^{2} \)
71 \( 1 + (3.05 + 9.40i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-3.87 - 2.81i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-4.58 - 1.49i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (3.20 + 9.86i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 1.52iT - 89T^{2} \)
97 \( 1 + (9.72 + 3.16i)T + (78.4 + 57.0i)T^{2} \)
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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

−13.78710450421588609835894835735, −12.83411613794866975352317971922, −11.60898919816604253508759493740, −10.67571230203791250706774598351, −9.618912787492999438473276409313, −8.173186740300095612866476262260, −7.23851940909149477480736067132, −5.70806072397227831698155665232, −3.54028921763348353903339987533, −0.48443615692537183488730654938, 3.80629814589997344397009653711, 5.17553070318209467933068569131, 7.20194396903538933053385996919, 8.121138820891291277767293502604, 9.509464793180659207709715782528, 10.13973794133921283014029761056, 11.69570297585521357909754560705, 12.58475096870131308004794177516, 13.86917684873421051271042283298, 15.72498685669546264936068584454

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