↑

Properties

Label 1-3149-3149.1058-r1-0-0
Degree $1$
Conductor $3149$
Sign $-0.194 + 0.980i$
Analytic cond. $338.407$
Root an. cond. $338.407$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.426 − 0.904i)2-s + (0.616 − 0.787i)3-s + (−0.635 − 0.771i)4-s + (−0.922 − 0.386i)5-s + (−0.449 − 0.893i)6-s + (−0.545 + 0.837i)7-s + (−0.969 + 0.245i)8-s + (−0.239 − 0.970i)9-s + (−0.743 + 0.668i)10-s + (0.215 + 0.976i)11-s + (−0.999 + 0.0248i)12-s + (0.791 + 0.611i)13-s + (0.524 + 0.851i)14-s + (−0.873 + 0.487i)15-s + (−0.191 + 0.981i)16-s + (0.997 + 0.0744i)17-s + ⋯
L(s)  = 1  + (0.426 − 0.904i)2-s + (0.616 − 0.787i)3-s + (−0.635 − 0.771i)4-s + (−0.922 − 0.386i)5-s + (−0.449 − 0.893i)6-s + (−0.545 + 0.837i)7-s + (−0.969 + 0.245i)8-s + (−0.239 − 0.970i)9-s + (−0.743 + 0.668i)10-s + (0.215 + 0.976i)11-s + (−0.999 + 0.0248i)12-s + (0.791 + 0.611i)13-s + (0.524 + 0.851i)14-s + (−0.873 + 0.487i)15-s + (−0.191 + 0.981i)16-s + (0.997 + 0.0744i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3149 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.194 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3149 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.194 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3149\)    =    \(47 \cdot 67\)
Sign: $-0.194 + 0.980i$
Analytic conductor: \(338.407\)
Root analytic conductor: \(338.407\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3149} (1058, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3149,\ (1:\ ),\ -0.194 + 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.001212401363 + 0.001476820521i\)
\(L(\frac12)\) \(\approx\) \(0.001212401363 + 0.001476820521i\)
\(L(1)\) \(\approx\) \(0.9034839477 - 0.6246339882i\)
\(L(1)\) \(\approx\) \(0.9034839477 - 0.6246339882i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad47 \( 1 \)
67 \( 1 \)
good2 \( 1 + (0.426 - 0.904i)T \)
3 \( 1 + (0.616 - 0.787i)T \)
5 \( 1 + (-0.922 - 0.386i)T \)
7 \( 1 + (-0.545 + 0.837i)T \)
11 \( 1 + (0.215 + 0.976i)T \)
13 \( 1 + (0.791 + 0.611i)T \)
17 \( 1 + (0.997 + 0.0744i)T \)
19 \( 1 + (0.984 + 0.172i)T \)
23 \( 1 + (0.154 + 0.987i)T \)
29 \( 1 + (-0.334 + 0.942i)T \)
31 \( 1 + (-0.867 + 0.498i)T \)
37 \( 1 + (-0.917 - 0.398i)T \)
41 \( 1 + (0.820 - 0.571i)T \)
43 \( 1 + (-0.392 - 0.919i)T \)
53 \( 1 + (-0.626 + 0.779i)T \)
59 \( 1 + (0.437 + 0.899i)T \)
61 \( 1 + (-0.992 + 0.123i)T \)
71 \( 1 + (0.130 - 0.991i)T \)
73 \( 1 + (0.323 + 0.946i)T \)
79 \( 1 + (0.358 + 0.933i)T \)
83 \( 1 + (-0.215 - 0.976i)T \)
89 \( 1 + (-0.896 + 0.443i)T \)
97 \( 1 + (-0.460 - 0.887i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.788727787874977373876239707521, −17.78650804004061422072373720306, −16.63107471367901650098315214659, −16.41519819840430274154147588138, −15.86729492701009505645452618446, −15.1461644280040181207453296687, −14.47701140986874377017846023217, −13.902506936867412012020308292255, −13.30827352477243585231264515931, −12.49444817235771751390286512186, −11.408762055980180141196621268327, −10.87641367049897767030483507493, −9.92749662061487270041253464980, −9.26337750541399520890660414038, −8.195960185201520386987993050580, −8.011812084228542743192306715713, −7.1793415072762097994444834495, −6.30458381754268594634853334396, −5.51821540603342519944261133398, −4.58837491396718129764223776747, −3.80523055137092595834015354212, −3.35534343844520573696318278764, −2.902365232432199900905070658983, −0.88341477578417802955832053277, −0.00028207837361178106898766644, 1.29295901938861764845554359663, 1.63360755147265075272543457555, 2.78620170962412503494286900302, 3.53008195926389080888167519684, 3.90671685478064622442215291521, 5.193769568241796983820628567522, 5.73529493406983249246341236670, 6.87403999369101144338531810866, 7.48065164742067675208469853146, 8.474065122127206692727468538426, 9.208119432100511255570103724410, 9.44915286724149507685205154756, 10.664798571111999929109652294994, 11.62239144264783483747218006759, 12.16327998590509055238546876676, 12.4564697217176891501572564628, 13.186103519582153209797270257245, 14.025512197570876508890114072912, 14.61197021852252027712819200998, 15.40247845637792199527002630776, 15.90109851247124918915777327730, 16.99470350456861787052876323750, 18.212581639331691671654575508720, 18.35816295915950518726957683291, 19.25185344109344320247789344503

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