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Properties

Label 2-1776-37.10-c1-0-11
Degree $2$
Conductor $1776$
Sign $0.998 - 0.0575i$
Analytic cond. $14.1814$
Root an. cond. $3.76582$
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  + (−0.5 − 0.866i)3-s + (1.18 + 2.05i)5-s + (−0.686 − 1.18i)7-s + (−0.499 + 0.866i)9-s − 2·11-s + (−0.686 − 1.18i)13-s + (1.18 − 2.05i)15-s + (0.813 − 1.40i)17-s + (2.37 + 4.10i)19-s + (−0.686 + 1.18i)21-s + (−0.313 + 0.543i)25-s + 0.999·27-s + 4.37·29-s + 9.37·31-s + (1 + 1.73i)33-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (0.530 + 0.918i)5-s + (−0.259 − 0.449i)7-s + (−0.166 + 0.288i)9-s − 0.603·11-s + (−0.190 − 0.329i)13-s + (0.306 − 0.530i)15-s + (0.197 − 0.341i)17-s + (0.544 + 0.942i)19-s + (−0.149 + 0.259i)21-s + (−0.0627 + 0.108i)25-s + 0.192·27-s + 0.811·29-s + 1.68·31-s + (0.174 + 0.301i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0575i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0575i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1776\)    =    \(2^{4} \cdot 3 \cdot 37\)
Sign: $0.998 - 0.0575i$
Analytic conductor: \(14.1814\)
Root analytic conductor: \(3.76582\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1776} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1776,\ (\ :1/2),\ 0.998 - 0.0575i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.596022641\)
\(L(\frac12)\) \(\approx\) \(1.596022641\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (2.55 + 5.51i)T \)
good5 \( 1 + (-1.18 - 2.05i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.686 + 1.18i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + (0.686 + 1.18i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.813 + 1.40i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.37 - 4.10i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 4.37T + 29T^{2} \)
31 \( 1 - 9.37T + 31T^{2} \)
41 \( 1 + (-2.18 - 3.78i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 9.37T + 43T^{2} \)
47 \( 1 + 2T + 47T^{2} \)
53 \( 1 + (5.74 - 9.94i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.55 - 7.89i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.05 - 3.56i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.74 - 9.94i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 2.62T + 73T^{2} \)
79 \( 1 + (-0.686 - 1.18i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4 + 6.92i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.81 - 3.14i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 5.74T + 97T^{2} \)
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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

−9.498220184559268129157390305763, −8.299704690472883913952096828337, −7.58692712983768304064173923539, −6.92041090967946691048689898523, −6.13274972580929272584536475965, −5.48428217619961894852148899885, −4.34810281474110168221928592375, −3.08020370733812806704873635716, −2.42094513512557689561937672309, −0.943361653793592245379568385408, 0.837062962525774550714951484138, 2.29074538990447492118468474435, 3.32393753390828429848262397647, 4.69830644843804125990618541642, 5.01733627681493050702969111726, 5.96254136277841905348173468431, 6.69859056591987995778819803773, 7.891656430595571616425310116568, 8.652230981149558136938545597623, 9.372620670572184831860315950479

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