Properties

Label 1-569-569.37-r1-0-0
Degree $1$
Conductor $569$
Sign $0.570 + 0.821i$
Analytic cond. $61.1475$
Root an. cond. $61.1475$
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  + (−0.873 − 0.487i)2-s + (−0.0993 + 0.995i)3-s + (0.525 + 0.850i)4-s + (−0.367 − 0.930i)5-s + (0.571 − 0.820i)6-s + (0.993 + 0.110i)7-s + (−0.0442 − 0.999i)8-s + (−0.980 − 0.197i)9-s + (−0.132 + 0.991i)10-s + (0.999 + 0.0110i)11-s + (−0.898 + 0.438i)12-s + (0.801 + 0.598i)13-s + (−0.814 − 0.580i)14-s + (0.962 − 0.273i)15-s + (−0.448 + 0.894i)16-s + (0.988 − 0.154i)17-s + ⋯
L(s)  = 1  + (−0.873 − 0.487i)2-s + (−0.0993 + 0.995i)3-s + (0.525 + 0.850i)4-s + (−0.367 − 0.930i)5-s + (0.571 − 0.820i)6-s + (0.993 + 0.110i)7-s + (−0.0442 − 0.999i)8-s + (−0.980 − 0.197i)9-s + (−0.132 + 0.991i)10-s + (0.999 + 0.0110i)11-s + (−0.898 + 0.438i)12-s + (0.801 + 0.598i)13-s + (−0.814 − 0.580i)14-s + (0.962 − 0.273i)15-s + (−0.448 + 0.894i)16-s + (0.988 − 0.154i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.570 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 569 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.570 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(569\)
Sign: $0.570 + 0.821i$
Analytic conductor: \(61.1475\)
Root analytic conductor: \(61.1475\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{569} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 569,\ (1:\ ),\ 0.570 + 0.821i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.179242116 + 0.6163814776i\)
\(L(\frac12)\) \(\approx\) \(1.179242116 + 0.6163814776i\)
\(L(1)\) \(\approx\) \(0.7980094376 + 0.08924370463i\)
\(L(1)\) \(\approx\) \(0.7980094376 + 0.08924370463i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad569 \( 1 \)
good2 \( 1 + (-0.873 - 0.487i)T \)
3 \( 1 + (-0.0993 + 0.995i)T \)
5 \( 1 + (-0.367 - 0.930i)T \)
7 \( 1 + (0.993 + 0.110i)T \)
11 \( 1 + (0.999 + 0.0110i)T \)
13 \( 1 + (0.801 + 0.598i)T \)
17 \( 1 + (0.988 - 0.154i)T \)
19 \( 1 + (-0.973 + 0.230i)T \)
23 \( 1 + (-0.977 + 0.208i)T \)
29 \( 1 + (0.589 + 0.807i)T \)
31 \( 1 + (-0.766 + 0.641i)T \)
37 \( 1 + (0.624 + 0.780i)T \)
41 \( 1 + (0.839 + 0.544i)T \)
43 \( 1 + (-0.487 - 0.873i)T \)
47 \( 1 + (0.0773 - 0.997i)T \)
53 \( 1 + (-0.820 - 0.571i)T \)
59 \( 1 + (0.997 + 0.0773i)T \)
61 \( 1 + (0.428 - 0.903i)T \)
67 \( 1 + (0.988 + 0.154i)T \)
71 \( 1 + (0.0442 - 0.999i)T \)
73 \( 1 + (0.230 + 0.973i)T \)
79 \( 1 + (0.346 - 0.937i)T \)
83 \( 1 + (-0.845 + 0.534i)T \)
89 \( 1 + (-0.641 + 0.766i)T \)
97 \( 1 + (-0.0331 + 0.999i)T \)
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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

−23.29737949821407541090133614097, −22.41254798505969726383172434572, −21.06500846062783495501990073880, −20.022875555892402678771475897652, −19.33813544697348222141660668124, −18.625558790241157202348732240391, −17.90073560648350910328224964307, −17.38810441042840712923158545181, −16.41937127375543794165294053212, −15.229023035469819626663803028546, −14.42475715316030393295260060749, −14.04717394523162845924692801174, −12.52684742231914146167773471981, −11.39082293685550303084396716652, −11.10932391650494268170822745472, −10.0013660240588119208532376293, −8.63087879137719252364049760913, −7.962680264957763174298278120418, −7.350877037777464493737093497054, −6.28555634704296815019757921461, −5.79541928973646161162951041805, −4.085467231393492976293845422543, −2.57764907035949779220131932083, −1.58110902495236815062443506559, −0.56409390845118360099265447819, 0.95975806245435644914872010220, 1.89303857621579153445869841899, 3.60055514703925538074910060418, 4.149579370627990854202472146557, 5.24089013131491537955661516465, 6.50883455958527307744437279747, 8.01551872980404020929119498752, 8.56884348849510418951310290345, 9.24974747955311536070060269606, 10.18771119275073279234554762053, 11.221996821214332471749962980401, 11.73662959657306794292708947178, 12.508615768804459805462018519931, 14.00007029994922689172853837133, 14.88577661199203186023179513111, 15.98344585062322394494465027927, 16.55957449568214235383702997960, 17.1704204644009656608048997909, 18.06344262666388828904746119802, 19.14673854106870719970566337596, 20.09343917253755517975305300935, 20.553171522053279864370595661226, 21.44067431709332220787880055548, 21.78335807261068113268187131944, 23.23904002087107636872851901042

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