↑

Properties

Label 1-2057-2057.70-r0-0-0
Degree $1$
Conductor $2057$
Sign $0.756 - 0.653i$
Analytic cond. $9.55266$
Root an. cond. $9.55266$
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  + (0.999 − 0.0285i)2-s + (−0.987 + 0.156i)3-s + (0.998 − 0.0570i)4-s + (−0.970 − 0.240i)5-s + (−0.982 + 0.184i)6-s + (0.552 − 0.833i)7-s + (0.996 − 0.0855i)8-s + (0.951 − 0.309i)9-s + (−0.977 − 0.212i)10-s + (−0.977 + 0.212i)12-s + (0.254 − 0.967i)13-s + (0.528 − 0.848i)14-s + (0.996 + 0.0855i)15-s + (0.993 − 0.113i)16-s + (0.941 − 0.336i)18-s + (0.717 + 0.696i)19-s + ⋯
L(s)  = 1  + (0.999 − 0.0285i)2-s + (−0.987 + 0.156i)3-s + (0.998 − 0.0570i)4-s + (−0.970 − 0.240i)5-s + (−0.982 + 0.184i)6-s + (0.552 − 0.833i)7-s + (0.996 − 0.0855i)8-s + (0.951 − 0.309i)9-s + (−0.977 − 0.212i)10-s + (−0.977 + 0.212i)12-s + (0.254 − 0.967i)13-s + (0.528 − 0.848i)14-s + (0.996 + 0.0855i)15-s + (0.993 − 0.113i)16-s + (0.941 − 0.336i)18-s + (0.717 + 0.696i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2057 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.756 - 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2057 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.756 - 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2057\)    =    \(11^{2} \cdot 17\)
Sign: $0.756 - 0.653i$
Analytic conductor: \(9.55266\)
Root analytic conductor: \(9.55266\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2057} (70, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2057,\ (0:\ ),\ 0.756 - 0.653i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.155701751 - 0.8021911509i\)
\(L(\frac12)\) \(\approx\) \(2.155701751 - 0.8021911509i\)
\(L(1)\) \(\approx\) \(1.465901666 - 0.2318188032i\)
\(L(1)\) \(\approx\) \(1.465901666 - 0.2318188032i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.999 - 0.0285i)T \)
3 \( 1 + (-0.987 + 0.156i)T \)
5 \( 1 + (-0.970 - 0.240i)T \)
7 \( 1 + (0.552 - 0.833i)T \)
13 \( 1 + (0.254 - 0.967i)T \)
19 \( 1 + (0.717 + 0.696i)T \)
23 \( 1 + (-0.349 + 0.936i)T \)
29 \( 1 + (0.955 + 0.295i)T \)
31 \( 1 + (0.995 - 0.0997i)T \)
37 \( 1 + (0.402 + 0.915i)T \)
41 \( 1 + (0.128 - 0.991i)T \)
43 \( 1 + (-0.755 + 0.654i)T \)
47 \( 1 + (-0.941 - 0.336i)T \)
53 \( 1 + (-0.113 + 0.993i)T \)
59 \( 1 + (-0.791 - 0.610i)T \)
61 \( 1 + (0.727 - 0.686i)T \)
67 \( 1 + (-0.959 - 0.281i)T \)
71 \( 1 + (0.946 - 0.322i)T \)
73 \( 1 + (0.963 - 0.268i)T \)
79 \( 1 + (-0.376 - 0.926i)T \)
83 \( 1 + (0.676 - 0.736i)T \)
89 \( 1 + (0.142 + 0.989i)T \)
97 \( 1 + (0.240 + 0.970i)T \)
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

−19.949492648078651157296919759087, −19.29408520630076157797383596714, −18.4982101894413312267500592055, −17.87620733634688395695433395624, −16.82407889817341276262571381465, −16.062490286292135107303792277255, −15.73129725399865665943558684505, −14.89327051307959914183757529544, −14.19022517424139450048013881924, −13.28776707265157840976131016054, −12.35959671167656540659813793151, −11.88034325421157643259202945624, −11.417802360924639796287129239501, −10.83741358552475360583687328120, −9.82058664711713085724569632473, −8.49444329059832716898562500171, −7.78342967772282737701409700324, −6.80257226114909518952174030601, −6.41506881171046326285580323173, −5.406063389695140482978909271585, −4.637391305256987818805229175289, −4.21197593636214202288595409384, −3.00458548006497890698925920749, −2.12399075279247235484588535864, −1.00524519111662420261147863017, 0.81715672368303694360119733351, 1.53810839401514134013619551692, 3.215980679249919933611632291814, 3.74464016152745694295139248279, 4.69073904541834643134953196417, 5.05928520599464392816859063762, 6.042494737010572733285764666221, 6.840254964325198534214341186133, 7.73277319881936385572332356077, 8.087441290802836587314767171537, 9.79997955722320119767121987308, 10.52089545156086549599468857632, 11.11335809547141494230523466469, 11.84959411757114089712790890453, 12.245763003989711690669564614244, 13.18212239098759951223973794719, 13.82681782865849281910905085408, 14.82440017445451808035040837566, 15.52858891853488680773436042680, 16.02060755235203095619260330012, 16.74426527211580779138471020029, 17.43198704859257402978868527807, 18.258662749296425757623832867991, 19.2804045820740578570335544924, 20.09360332603281510901918034847

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