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Properties

Label 1-2057-2057.150-r0-0-0
Degree $1$
Conductor $2057$
Sign $-0.914 + 0.404i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.982 + 0.184i)2-s + (0.522 + 0.852i)3-s + (0.931 + 0.362i)4-s + (0.00713 + 0.999i)5-s + (0.356 + 0.934i)6-s + (−0.121 + 0.992i)7-s + (0.848 + 0.528i)8-s + (−0.453 + 0.891i)9-s + (−0.177 + 0.984i)10-s + (0.177 + 0.984i)12-s + (0.633 − 0.774i)13-s + (−0.302 + 0.953i)14-s + (−0.848 + 0.528i)15-s + (0.736 + 0.676i)16-s + (−0.610 + 0.791i)18-s + (−0.295 − 0.955i)19-s + ⋯
L(s)  = 1  + (0.982 + 0.184i)2-s + (0.522 + 0.852i)3-s + (0.931 + 0.362i)4-s + (0.00713 + 0.999i)5-s + (0.356 + 0.934i)6-s + (−0.121 + 0.992i)7-s + (0.848 + 0.528i)8-s + (−0.453 + 0.891i)9-s + (−0.177 + 0.984i)10-s + (0.177 + 0.984i)12-s + (0.633 − 0.774i)13-s + (−0.302 + 0.953i)14-s + (−0.848 + 0.528i)15-s + (0.736 + 0.676i)16-s + (−0.610 + 0.791i)18-s + (−0.295 − 0.955i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2057 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.914 + 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2057 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.914 + 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2057\)    =    \(11^{2} \cdot 17\)
Sign: $-0.914 + 0.404i$
Analytic conductor: \(9.55266\)
Root analytic conductor: \(9.55266\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2057} (150, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2057,\ (0:\ ),\ -0.914 + 0.404i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7888610644 + 3.735723339i\)
\(L(\frac12)\) \(\approx\) \(0.7888610644 + 3.735723339i\)
\(L(1)\) \(\approx\) \(1.644522649 + 1.580706948i\)
\(L(1)\) \(\approx\) \(1.644522649 + 1.580706948i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.982 + 0.184i)T \)
3 \( 1 + (0.522 + 0.852i)T \)
5 \( 1 + (0.00713 + 0.999i)T \)
7 \( 1 + (-0.121 + 0.992i)T \)
13 \( 1 + (0.633 - 0.774i)T \)
19 \( 1 + (-0.295 - 0.955i)T \)
23 \( 1 + (-0.0356 + 0.999i)T \)
29 \( 1 + (0.929 - 0.369i)T \)
31 \( 1 + (-0.604 + 0.796i)T \)
37 \( 1 + (0.944 + 0.329i)T \)
41 \( 1 + (-0.0499 - 0.998i)T \)
43 \( 1 + (0.997 - 0.0713i)T \)
47 \( 1 + (-0.791 + 0.610i)T \)
53 \( 1 + (-0.0428 + 0.999i)T \)
59 \( 1 + (-0.903 - 0.428i)T \)
61 \( 1 + (-0.978 - 0.205i)T \)
67 \( 1 + (0.959 - 0.281i)T \)
71 \( 1 + (0.845 + 0.534i)T \)
73 \( 1 + (-0.981 - 0.191i)T \)
79 \( 1 + (-0.149 + 0.988i)T \)
83 \( 1 + (0.621 - 0.783i)T \)
89 \( 1 + (0.989 + 0.142i)T \)
97 \( 1 + (0.712 - 0.702i)T \)
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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

−19.9407976155225397463809129056, −19.071130748332875560812613391374, −18.325450006363220294600021540279, −17.22427085626790566999046112593, −16.46436064202764778952017257477, −16.10286254450885597489059017755, −14.796979943654747176844907551053, −14.3186421903381020253254317953, −13.54255343743575257718565764305, −13.08803964467876400396106656705, −12.452633940536024965404320930606, −11.76094128996903228272919937304, −10.928335051143315036742887066138, −9.96313570325618516798747972482, −9.07740701611561557059445327011, −8.10372265022011075016329283523, −7.55114941104789374169335859153, −6.45815631351362799142064688444, −6.14716281716048369639763915778, −4.88806359898672471400098726886, −4.10947171220529034864032841273, −3.54610563308006754637463149260, −2.35837976841078681327327994114, −1.526955018900607302055858897740, −0.82308940406265380300261040050, 1.86317585780093479879342120762, 2.83749581037435970647625541488, 3.08678614186657953284368751636, 4.02209381722931578700130289991, 4.95487763100615359236881989874, 5.74172841985534394951729717055, 6.345765592384263825147403004, 7.41001130171426368514900497891, 8.137142753684304248414110698640, 9.02250964134254815121273178686, 9.92341092116934336963949632922, 10.88339121680434912449648799963, 11.16599908774868731091506143299, 12.1651035201562949531794710687, 13.06645298536002089222822790058, 13.834754038022738907786196956048, 14.406738940721449814542169364394, 15.282632568252872671544711864226, 15.512943491850665320474865174942, 16.03698169105295923073636241475, 17.23905727978520750901571452363, 17.95015614835935519441600369916, 18.98695137520120281461866629685, 19.64511946013661648755262431703, 20.30309667553735560132732152938

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