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Properties

Label 1-2057-2057.584-r1-0-0
Degree $1$
Conductor $2057$
Sign $0.921 + 0.387i$
Analytic cond. $221.055$
Root an. cond. $221.055$
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  + (−0.997 + 0.0713i)2-s + (0.923 − 0.382i)3-s + (0.989 − 0.142i)4-s + (−0.821 − 0.570i)5-s + (−0.894 + 0.447i)6-s + (0.627 − 0.778i)7-s + (−0.977 + 0.212i)8-s + (0.707 − 0.707i)9-s + (0.860 + 0.510i)10-s + (0.860 − 0.510i)12-s + (−0.989 + 0.142i)13-s + (−0.570 + 0.821i)14-s + (−0.977 − 0.212i)15-s + (0.959 − 0.281i)16-s + (−0.654 + 0.755i)18-s + (−0.349 + 0.936i)19-s + ⋯
L(s)  = 1  + (−0.997 + 0.0713i)2-s + (0.923 − 0.382i)3-s + (0.989 − 0.142i)4-s + (−0.821 − 0.570i)5-s + (−0.894 + 0.447i)6-s + (0.627 − 0.778i)7-s + (−0.977 + 0.212i)8-s + (0.707 − 0.707i)9-s + (0.860 + 0.510i)10-s + (0.860 − 0.510i)12-s + (−0.989 + 0.142i)13-s + (−0.570 + 0.821i)14-s + (−0.977 − 0.212i)15-s + (0.959 − 0.281i)16-s + (−0.654 + 0.755i)18-s + (−0.349 + 0.936i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2057 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.921 + 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2057 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.921 + 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2057\)    =    \(11^{2} \cdot 17\)
Sign: $0.921 + 0.387i$
Analytic conductor: \(221.055\)
Root analytic conductor: \(221.055\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2057} (584, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2057,\ (1:\ ),\ 0.921 + 0.387i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9671712879 + 0.1950200219i\)
\(L(\frac12)\) \(\approx\) \(0.9671712879 + 0.1950200219i\)
\(L(1)\) \(\approx\) \(0.7521357554 - 0.1908972197i\)
\(L(1)\) \(\approx\) \(0.7521357554 - 0.1908972197i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.997 + 0.0713i)T \)
3 \( 1 + (0.923 - 0.382i)T \)
5 \( 1 + (-0.821 - 0.570i)T \)
7 \( 1 + (0.627 - 0.778i)T \)
13 \( 1 + (-0.989 + 0.142i)T \)
19 \( 1 + (-0.349 + 0.936i)T \)
23 \( 1 + (-0.994 - 0.106i)T \)
29 \( 1 + (-0.681 + 0.731i)T \)
31 \( 1 + (-0.247 - 0.968i)T \)
37 \( 1 + (0.247 + 0.968i)T \)
41 \( 1 + (-0.447 - 0.894i)T \)
43 \( 1 + (0.977 - 0.212i)T \)
47 \( 1 + (-0.755 + 0.654i)T \)
53 \( 1 + (-0.479 - 0.877i)T \)
59 \( 1 + (0.997 + 0.0713i)T \)
61 \( 1 + (-0.948 + 0.315i)T \)
67 \( 1 + (0.654 - 0.755i)T \)
71 \( 1 + (0.731 + 0.681i)T \)
73 \( 1 + (0.627 + 0.778i)T \)
79 \( 1 + (-0.984 + 0.177i)T \)
83 \( 1 + (0.479 + 0.877i)T \)
89 \( 1 + (0.909 + 0.415i)T \)
97 \( 1 + (-0.177 + 0.984i)T \)
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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

−19.62259831824679797821585268603, −19.07413249656438093435627438942, −18.286946511808286447908135009252, −17.74575596394740739145829528497, −16.722815414848187961355862894112, −15.81299172213071882566004297142, −15.41945821793441777009964251810, −14.73619408298964934802211531614, −14.256525120010007993869500115142, −12.8782729397367108826815583561, −12.09340168169758456168555968939, −11.36222306907762978121967536944, −10.69830486017652689000648802990, −9.87687001452741510949361042358, −9.15210009498688268524553076714, −8.45690190947267496069131947100, −7.75349897655998248489899007399, −7.33055889264351534272353933704, −6.29059224819280519950595569583, −5.07355825525907783332673613617, −4.13642916190837591856030851027, −3.12488049707608761408346274425, −2.46098347001435526818630130136, −1.80360477172811274016804244170, −0.270525665979905315357114707570, 0.70581067887226181678482021713, 1.59727058236900736698798533652, 2.310933867694664223931873896544, 3.544599179852216544658866591737, 4.16574695178024307843770443194, 5.292829151008711012062513613230, 6.58763792292548647041865603327, 7.33353088733884911641942863843, 7.9660196172575716720470091049, 8.243290368126006771611692407251, 9.26399676650713609463192939160, 9.880269189867426822523860226075, 10.74949289865771695101809716922, 11.66942028789208574141666426886, 12.289744877625254746111497799105, 13.02283418064821225868320003587, 14.206710720986610848853465537586, 14.68633034101331377563704978030, 15.39944329193543851394712055625, 16.26975905240064701028127671999, 16.89737687284085013037943782175, 17.57552263670444545246926626420, 18.51031894352988715956702288967, 19.057836838892090245555150257899, 19.74703177425806639957360618133

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