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Properties

Label 2-59e2-59.47-c0-0-2
Degree $2$
Conductor $3481$
Sign $0.788 + 0.615i$
Analytic cond. $1.73724$
Root an. cond. $1.31804$
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  + (0.856 − 0.515i)3-s + (0.796 − 0.605i)4-s + (0.370 + 0.928i)5-s + (−0.907 − 0.419i)7-s + (0.370 − 0.928i)12-s + (0.796 + 0.605i)15-s + (0.267 − 0.963i)16-s + (1.81 − 0.839i)17-s + (−0.976 − 0.214i)19-s + (0.856 + 0.515i)20-s + (−0.994 + 0.108i)21-s + (0.0541 + 0.998i)27-s + (−0.976 + 0.214i)28-s + (0.947 + 0.319i)29-s + (0.0541 − 0.998i)35-s + ⋯
L(s)  = 1  + (0.856 − 0.515i)3-s + (0.796 − 0.605i)4-s + (0.370 + 0.928i)5-s + (−0.907 − 0.419i)7-s + (0.370 − 0.928i)12-s + (0.796 + 0.605i)15-s + (0.267 − 0.963i)16-s + (1.81 − 0.839i)17-s + (−0.976 − 0.214i)19-s + (0.856 + 0.515i)20-s + (−0.994 + 0.108i)21-s + (0.0541 + 0.998i)27-s + (−0.976 + 0.214i)28-s + (0.947 + 0.319i)29-s + (0.0541 − 0.998i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3481 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3481 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3481\)    =    \(59^{2}\)
Sign: $0.788 + 0.615i$
Analytic conductor: \(1.73724\)
Root analytic conductor: \(1.31804\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3481} (1404, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3481,\ (\ :0),\ 0.788 + 0.615i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.040846846\)
\(L(\frac12)\) \(\approx\) \(2.040846846\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad59 \( 1 \)
good2 \( 1 + (-0.796 + 0.605i)T^{2} \)
3 \( 1 + (-0.856 + 0.515i)T + (0.468 - 0.883i)T^{2} \)
5 \( 1 + (-0.370 - 0.928i)T + (-0.725 + 0.687i)T^{2} \)
7 \( 1 + (0.907 + 0.419i)T + (0.647 + 0.762i)T^{2} \)
11 \( 1 + (0.856 - 0.515i)T^{2} \)
13 \( 1 + (0.561 - 0.827i)T^{2} \)
17 \( 1 + (-1.81 + 0.839i)T + (0.647 - 0.762i)T^{2} \)
19 \( 1 + (0.976 + 0.214i)T + (0.907 + 0.419i)T^{2} \)
23 \( 1 + (0.947 - 0.319i)T^{2} \)
29 \( 1 + (-0.947 - 0.319i)T + (0.796 + 0.605i)T^{2} \)
31 \( 1 + (-0.907 + 0.419i)T^{2} \)
37 \( 1 + (0.370 - 0.928i)T^{2} \)
41 \( 1 + (-0.161 + 0.986i)T + (-0.947 - 0.319i)T^{2} \)
43 \( 1 + (0.856 + 0.515i)T^{2} \)
47 \( 1 + (0.725 + 0.687i)T^{2} \)
53 \( 1 + (0.647 - 0.762i)T + (-0.161 - 0.986i)T^{2} \)
61 \( 1 + (-0.796 + 0.605i)T^{2} \)
67 \( 1 + (0.370 + 0.928i)T^{2} \)
71 \( 1 + (0.740 - 1.85i)T + (-0.725 - 0.687i)T^{2} \)
73 \( 1 + (-0.976 - 0.214i)T^{2} \)
79 \( 1 + (-0.856 - 0.515i)T + (0.468 + 0.883i)T^{2} \)
83 \( 1 + (0.994 + 0.108i)T^{2} \)
89 \( 1 + (-0.796 - 0.605i)T^{2} \)
97 \( 1 + (-0.976 + 0.214i)T^{2} \)
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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

−8.636178737525638803605104218033, −7.69759690105514180611130462414, −7.15640193888811987894801220823, −6.62697293484671982571019152545, −5.93755069622971652319630996656, −5.06396468110270187132698347273, −3.59885861831713196136895684300, −2.84204391043889193509780145285, −2.43825997441742561570611971961, −1.21161325269114636156598341027, 1.47580055313194902918023304104, 2.60921219717682046087662406933, 3.29229356729212952654257561253, 3.91786474618019883246712631194, 4.98347192634958889744359784770, 6.15221908258803213253937248067, 6.31020644857140461005788742466, 7.65308347640806051623473508206, 8.229500187897644961523627263388, 8.765962291471581743134167858192

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