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Properties

Label 1-1312-1312.379-r1-0-0
Degree $1$
Conductor $1312$
Sign $0.643 - 0.765i$
Analytic cond. $140.993$
Root an. cond. $140.993$
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  + (0.707 − 0.707i)3-s + (−0.987 − 0.156i)5-s + (−0.951 − 0.309i)7-s − i·9-s + (−0.156 − 0.987i)11-s + (0.891 + 0.453i)13-s + (−0.809 + 0.587i)15-s + (0.809 + 0.587i)17-s + (0.453 + 0.891i)19-s + (−0.891 + 0.453i)21-s + (−0.951 + 0.309i)23-s + (0.951 + 0.309i)25-s + (−0.707 − 0.707i)27-s + (0.156 − 0.987i)29-s + (0.809 + 0.587i)31-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)3-s + (−0.987 − 0.156i)5-s + (−0.951 − 0.309i)7-s − i·9-s + (−0.156 − 0.987i)11-s + (0.891 + 0.453i)13-s + (−0.809 + 0.587i)15-s + (0.809 + 0.587i)17-s + (0.453 + 0.891i)19-s + (−0.891 + 0.453i)21-s + (−0.951 + 0.309i)23-s + (0.951 + 0.309i)25-s + (−0.707 − 0.707i)27-s + (0.156 − 0.987i)29-s + (0.809 + 0.587i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1312 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.643 - 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1312 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.643 - 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1312\)    =    \(2^{5} \cdot 41\)
Sign: $0.643 - 0.765i$
Analytic conductor: \(140.993\)
Root analytic conductor: \(140.993\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1312} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1312,\ (1:\ ),\ 0.643 - 0.765i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.759944916 - 0.8193753540i\)
\(L(\frac12)\) \(\approx\) \(1.759944916 - 0.8193753540i\)
\(L(1)\) \(\approx\) \(1.035343169 - 0.3256899177i\)
\(L(1)\) \(\approx\) \(1.035343169 - 0.3256899177i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
41 \( 1 \)
good3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (-0.987 - 0.156i)T \)
7 \( 1 + (-0.951 - 0.309i)T \)
11 \( 1 + (-0.156 - 0.987i)T \)
13 \( 1 + (0.891 + 0.453i)T \)
17 \( 1 + (0.809 + 0.587i)T \)
19 \( 1 + (0.453 + 0.891i)T \)
23 \( 1 + (-0.951 + 0.309i)T \)
29 \( 1 + (0.156 - 0.987i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (-0.987 - 0.156i)T \)
43 \( 1 + (-0.453 + 0.891i)T \)
47 \( 1 + (0.309 + 0.951i)T \)
53 \( 1 + (0.987 + 0.156i)T \)
59 \( 1 + (0.453 - 0.891i)T \)
61 \( 1 + (0.453 + 0.891i)T \)
67 \( 1 + (-0.156 + 0.987i)T \)
71 \( 1 + (-0.587 + 0.809i)T \)
73 \( 1 - iT \)
79 \( 1 + T \)
83 \( 1 + (-0.707 + 0.707i)T \)
89 \( 1 + (0.951 + 0.309i)T \)
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

−20.57727571899359415969857841704, −20.19072011090483413618814172192, −19.52795491249843516354682442436, −18.70946574110246904765219026579, −18.08325007056855250200703561870, −16.7458009900828510493278034359, −16.01307321379976746825372990020, −15.54651862415098967986104458447, −15.03440373613298920641307539965, −14.00929667180078972709905318716, −13.259603745834765070002019072992, −12.316292613788573925429232591980, −11.65007077622477392263330115006, −10.47562756458091917794494498041, −10.036430405021205510995052030767, −9.05685661321462814959519188853, −8.39763364301398902655929098823, −7.48837772155773139875829038327, −6.797632462891758635528985294758, −5.495029787373209223819351066805, −4.643117707057122294109340130735, −3.65237360252684774580333283968, −3.17486574958548957986669286250, −2.21449156199078165977557412603, −0.58178930859895088389242889904, 0.63108227002479418980613089161, 1.42164170166692332929495320639, 2.880567886585944853454196696339, 3.5929070922495526910819814953, 4.045392713113057121791793446914, 5.76257575141177459552421901651, 6.38203242397527403404036806621, 7.332251534378993314498526050785, 8.15053745619574384441411864634, 8.555296578818138586732380316153, 9.63201977063030157975412801860, 10.4803203796851856858706743596, 11.64066077928876238881271918216, 12.160771126500987159425676010861, 13.01101602004431467824078566606, 13.750880185827133246172496631827, 14.31520623011933655082048595112, 15.42115305314137764109043105616, 16.08588593880786344212767543218, 16.60363012088772367850855736691, 17.80297543414161399650240448421, 18.79878837067898744935162018689, 19.15690951229053032829056043342, 19.626656085361434807942969137006, 20.63135646781700842141958351149

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