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Properties

Label 1-40e2-1600.1309-r0-0-0
Degree $1$
Conductor $1600$
Sign $-0.532 + 0.846i$
Analytic cond. $7.43036$
Root an. cond. $7.43036$
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  + (0.972 − 0.233i)3-s + (−0.707 + 0.707i)7-s + (0.891 − 0.453i)9-s + (−0.649 − 0.760i)11-s + (−0.760 − 0.649i)13-s + (−0.587 + 0.809i)17-s + (−0.852 + 0.522i)19-s + (−0.522 + 0.852i)21-s + (−0.453 + 0.891i)23-s + (0.760 − 0.649i)27-s + (0.972 − 0.233i)29-s + (0.809 + 0.587i)31-s + (−0.809 − 0.587i)33-s + (−0.996 + 0.0784i)37-s + (−0.891 − 0.453i)39-s + ⋯
L(s)  = 1  + (0.972 − 0.233i)3-s + (−0.707 + 0.707i)7-s + (0.891 − 0.453i)9-s + (−0.649 − 0.760i)11-s + (−0.760 − 0.649i)13-s + (−0.587 + 0.809i)17-s + (−0.852 + 0.522i)19-s + (−0.522 + 0.852i)21-s + (−0.453 + 0.891i)23-s + (0.760 − 0.649i)27-s + (0.972 − 0.233i)29-s + (0.809 + 0.587i)31-s + (−0.809 − 0.587i)33-s + (−0.996 + 0.0784i)37-s + (−0.891 − 0.453i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.532 + 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.532 + 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.532 + 0.846i$
Analytic conductor: \(7.43036\)
Root analytic conductor: \(7.43036\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (1309, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1600,\ (0:\ ),\ -0.532 + 0.846i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3928808215 + 0.7113355420i\)
\(L(\frac12)\) \(\approx\) \(0.3928808215 + 0.7113355420i\)
\(L(1)\) \(\approx\) \(1.033859479 + 0.08738943876i\)
\(L(1)\) \(\approx\) \(1.033859479 + 0.08738943876i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.972 - 0.233i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
11 \( 1 + (-0.649 - 0.760i)T \)
13 \( 1 + (-0.760 - 0.649i)T \)
17 \( 1 + (-0.587 + 0.809i)T \)
19 \( 1 + (-0.852 + 0.522i)T \)
23 \( 1 + (-0.453 + 0.891i)T \)
29 \( 1 + (0.972 - 0.233i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (-0.996 + 0.0784i)T \)
41 \( 1 + (-0.891 + 0.453i)T \)
43 \( 1 + (-0.923 + 0.382i)T \)
47 \( 1 + (0.587 + 0.809i)T \)
53 \( 1 + (-0.522 + 0.852i)T \)
59 \( 1 + (-0.996 + 0.0784i)T \)
61 \( 1 + (-0.0784 + 0.996i)T \)
67 \( 1 + (0.522 + 0.852i)T \)
71 \( 1 + (-0.156 + 0.987i)T \)
73 \( 1 + (0.453 - 0.891i)T \)
79 \( 1 + (0.587 + 0.809i)T \)
83 \( 1 + (0.852 - 0.522i)T \)
89 \( 1 + (-0.891 - 0.453i)T \)
97 \( 1 + (-0.809 + 0.587i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.27080597985736259462008102300, −19.55010296207460375724738591567, −18.95411059849886242265509283880, −18.10318278744276856494279613633, −17.13674249547300289510460883055, −16.427538080390040478427277222491, −15.54582704564939125696469250538, −15.13840766922492722529370094633, −13.968910452239332049638679849715, −13.71765411061591254658121285005, −12.74626976954906433255957211768, −12.13903798275465543484654723891, −10.817759580561837917454139178011, −10.12675965286815771106362577206, −9.60646504289781462218558705684, −8.72749421428905433702600253796, −7.92680406127827470983880275407, −6.908054194569296135009979024087, −6.694026850466937123872434119957, −4.88222873802749019113518768368, −4.520509825818194734845640452282, −3.50363856202226744648802454357, −2.54507451905086750113991209972, −1.95925051015852249489924565837, −0.23488446546116974976782159958, 1.44299824236587383451448118113, 2.52226990891200548626294066217, 3.03491433001010466149632529403, 3.94180501837214163546785666787, 5.06461553456762057270011372049, 6.079619548677048259731687913850, 6.72683780112056580890799748958, 7.91207994801546104679918849804, 8.348924041431602919441146098860, 9.1150434723007094503318805014, 10.0645336512851441597351890163, 10.54104877654354642851871940774, 11.9378591059592167627390694007, 12.533664722851404835791226750449, 13.24504802312349386394806721763, 13.86120095066901991371132879612, 14.83907461424902713305969363262, 15.486463565370803369518416119830, 15.91339287536298020428659409779, 17.07431677585390824965314229261, 17.884106462075324010054634881880, 18.721674920680279036922448720165, 19.34907302651308392257175889126, 19.72453281500697118944392927243, 20.69142521757328646789558156527

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