↑

Properties

Label 1-293-293.42-r1-0-0
Degree $1$
Conductor $293$
Sign $-0.308 - 0.951i$
Analytic cond. $31.4872$
Root an. cond. $31.4872$
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  + (−0.128 + 0.991i)2-s + (−0.988 + 0.150i)3-s + (−0.966 − 0.255i)4-s + (0.337 − 0.941i)5-s + (−0.0215 − 0.999i)6-s + (−0.699 − 0.714i)7-s + (0.377 − 0.925i)8-s + (0.954 − 0.296i)9-s + (0.890 + 0.455i)10-s + (0.758 + 0.651i)11-s + (0.994 + 0.107i)12-s + (−0.0430 − 0.999i)13-s + (0.798 − 0.601i)14-s + (−0.192 + 0.981i)15-s + (0.869 + 0.493i)16-s + (−0.991 − 0.128i)17-s + ⋯
L(s)  = 1  + (−0.128 + 0.991i)2-s + (−0.988 + 0.150i)3-s + (−0.966 − 0.255i)4-s + (0.337 − 0.941i)5-s + (−0.0215 − 0.999i)6-s + (−0.699 − 0.714i)7-s + (0.377 − 0.925i)8-s + (0.954 − 0.296i)9-s + (0.890 + 0.455i)10-s + (0.758 + 0.651i)11-s + (0.994 + 0.107i)12-s + (−0.0430 − 0.999i)13-s + (0.798 − 0.601i)14-s + (−0.192 + 0.981i)15-s + (0.869 + 0.493i)16-s + (−0.991 − 0.128i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.308 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.308 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(293\)
Sign: $-0.308 - 0.951i$
Analytic conductor: \(31.4872\)
Root analytic conductor: \(31.4872\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{293} (42, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 293,\ (1:\ ),\ -0.308 - 0.951i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3347153999 - 0.4604166238i\)
\(L(\frac12)\) \(\approx\) \(0.3347153999 - 0.4604166238i\)
\(L(1)\) \(\approx\) \(0.6349420112 + 0.05531804837i\)
\(L(1)\) \(\approx\) \(0.6349420112 + 0.05531804837i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad293 \( 1 \)
good2 \( 1 + (-0.128 + 0.991i)T \)
3 \( 1 + (-0.988 + 0.150i)T \)
5 \( 1 + (0.337 - 0.941i)T \)
7 \( 1 + (-0.699 - 0.714i)T \)
11 \( 1 + (0.758 + 0.651i)T \)
13 \( 1 + (-0.0430 - 0.999i)T \)
17 \( 1 + (-0.991 - 0.128i)T \)
19 \( 1 + (0.981 + 0.192i)T \)
23 \( 1 + (0.999 + 0.0215i)T \)
29 \( 1 + (0.811 - 0.584i)T \)
31 \( 1 + (-0.941 + 0.337i)T \)
37 \( 1 + (-0.512 - 0.858i)T \)
41 \( 1 + (-0.729 - 0.683i)T \)
43 \( 1 + (0.397 + 0.917i)T \)
47 \( 1 + (0.668 - 0.744i)T \)
53 \( 1 + (0.651 - 0.758i)T \)
59 \( 1 + (-0.976 + 0.213i)T \)
61 \( 1 + (0.234 + 0.972i)T \)
67 \( 1 + (-0.908 - 0.417i)T \)
71 \( 1 + (0.847 - 0.530i)T \)
73 \( 1 + (0.966 - 0.255i)T \)
79 \( 1 + (-0.972 + 0.234i)T \)
83 \( 1 + (-0.436 - 0.899i)T \)
89 \( 1 + (0.972 + 0.234i)T \)
97 \( 1 + (-0.276 + 0.961i)T \)
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

−25.6846372087287816909007297167, −24.47136533879432768539593335790, −23.36795701519856455056856837452, −22.34974563495023066027823575420, −22.02399608937663412239889348753, −21.423954760307328771862797384, −19.87474952178700497016322548114, −18.828746488309636380113980960239, −18.58098768688227623628296944600, −17.496672120180576441727942634663, −16.666548786844772568755184946792, −15.46254271376529164782039075523, −14.09766865279755320841233700086, −13.296398243817824900123687834698, −12.17090238717371258960824985947, −11.43408038157003792946646768707, −10.76648479285233735986481750963, −9.63115204321513339298402874060, −8.90863212042992409784881622730, −7.06189727210725957805888197599, −6.26381549529896621321283448728, −5.124571587741290739181289159252, −3.7576393468661574211020770673, −2.62549486787547629477939147688, −1.36767647817398932343804845306, 0.24948926752907720656152052317, 1.19246929046356885206064563711, 3.83608652500793915068262466510, 4.797544697101728935835771211524, 5.631120832284113825103910016000, 6.6746145508407415279531609383, 7.45252444280922208193680433617, 8.96824161709596041820242627659, 9.720912607366186111371289107390, 10.60318377428652014352759128598, 12.18880096205173628696531121980, 12.9665941868079225350951171074, 13.744797268107085701266447634054, 15.20405388703672243825445318001, 16.01865504016292336845471991254, 16.74019537340636357565390389681, 17.47727992531478660822773661487, 17.999270264906921440939163166091, 19.507460151845175616751130052092, 20.369237548610871779588906191373, 21.7036551081224603663964072712, 22.73856399872600892691302130376, 22.98530774373545314182761100965, 24.209219340663298601630311931438, 24.81147841696579792290901446003

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