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Properties

Label 1-3643-3643.701-r1-0-0
Degree $1$
Conductor $3643$
Sign $-0.982 - 0.183i$
Analytic cond. $391.494$
Root an. cond. $391.494$
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  + (0.712 + 0.701i)2-s + (−0.909 + 0.414i)3-s + (0.0146 + 0.999i)4-s + (−0.979 − 0.202i)5-s + (−0.939 − 0.343i)6-s + (−0.926 − 0.375i)7-s + (−0.691 + 0.722i)8-s + (0.655 − 0.755i)9-s + (−0.555 − 0.831i)10-s + (−0.928 + 0.372i)11-s + (−0.428 − 0.903i)12-s + (−0.978 + 0.207i)13-s + (−0.396 − 0.917i)14-s + (0.974 − 0.222i)15-s + (−0.999 + 0.0293i)16-s + (0.167 + 0.985i)17-s + ⋯
L(s)  = 1  + (0.712 + 0.701i)2-s + (−0.909 + 0.414i)3-s + (0.0146 + 0.999i)4-s + (−0.979 − 0.202i)5-s + (−0.939 − 0.343i)6-s + (−0.926 − 0.375i)7-s + (−0.691 + 0.722i)8-s + (0.655 − 0.755i)9-s + (−0.555 − 0.831i)10-s + (−0.928 + 0.372i)11-s + (−0.428 − 0.903i)12-s + (−0.978 + 0.207i)13-s + (−0.396 − 0.917i)14-s + (0.974 − 0.222i)15-s + (−0.999 + 0.0293i)16-s + (0.167 + 0.985i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3643 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.982 - 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3643 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.982 - 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3643\)
Sign: $-0.982 - 0.183i$
Analytic conductor: \(391.494\)
Root analytic conductor: \(391.494\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3643} (701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3643,\ (1:\ ),\ -0.982 - 0.183i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.07582752611 + 0.8185435607i\)
\(L(\frac12)\) \(\approx\) \(-0.07582752611 + 0.8185435607i\)
\(L(1)\) \(\approx\) \(0.5367572549 + 0.4444583273i\)
\(L(1)\) \(\approx\) \(0.5367572549 + 0.4444583273i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3643 \( 1 \)
good2 \( 1 + (0.712 + 0.701i)T \)
3 \( 1 + (-0.909 + 0.414i)T \)
5 \( 1 + (-0.979 - 0.202i)T \)
7 \( 1 + (-0.926 - 0.375i)T \)
11 \( 1 + (-0.928 + 0.372i)T \)
13 \( 1 + (-0.978 + 0.207i)T \)
17 \( 1 + (0.167 + 0.985i)T \)
19 \( 1 + (-0.625 + 0.780i)T \)
23 \( 1 + (-0.456 + 0.889i)T \)
29 \( 1 + (0.772 - 0.635i)T \)
31 \( 1 + (-0.510 - 0.859i)T \)
37 \( 1 + (-0.989 + 0.144i)T \)
41 \( 1 + (0.204 - 0.978i)T \)
43 \( 1 + (0.887 - 0.459i)T \)
47 \( 1 + (-0.326 + 0.945i)T \)
53 \( 1 + (0.898 - 0.438i)T \)
59 \( 1 + (0.449 + 0.893i)T \)
61 \( 1 + (0.997 + 0.0706i)T \)
67 \( 1 + (0.611 + 0.791i)T \)
71 \( 1 + (-0.837 + 0.546i)T \)
73 \( 1 + (-0.696 + 0.717i)T \)
79 \( 1 + (0.859 - 0.511i)T \)
83 \( 1 + (0.443 + 0.896i)T \)
89 \( 1 + (0.143 - 0.989i)T \)
97 \( 1 + (-0.935 + 0.352i)T \)
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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

−18.34269717448190101278473206771, −17.739585393061243281824558123008, −16.3667267875009541103384331209, −16.137247461691344718538061722279, −15.45045452173271297505367392031, −14.68359189104074862275563916738, −13.795017926086539491828967595148, −12.98103285779905902895041313289, −12.478509823144403340856644710546, −12.060202601714275123636165592719, −11.3335245506845987421479771016, −10.56504555733024253887800679671, −10.21196601853250625031307259230, −9.17439845675679207032625641052, −8.205490061618175668189944807196, −7.108101161920524758323349108370, −6.78774460959480166359718906914, −5.87802658408660463431190598578, −4.98096923194258694177486912031, −4.70668273165781528483061547348, −3.51305983071884706302275492986, −2.78298597911767220100118670364, −2.206917950552643713608782957401, −0.64459089593555400799957854912, −0.30253005162702323864271451049, 0.528554349343678159198362075737, 2.16578711186731835978912346291, 3.28626042205412870824690455929, 4.07642716577138497952012761156, 4.315886561002486545108529860600, 5.4188797887066547052071658095, 5.84108612061239984237493897577, 6.832037328027825815750861947729, 7.34534913207478989671070366417, 8.00701247483555448654227733469, 8.95412923300475562459558398454, 9.992993535822485868781615993057, 10.45912217488351608055211629262, 11.47858260484856396721711068648, 12.1197020901260401163552401417, 12.67248534682708719735913212028, 13.05612638973703072324869541352, 14.193716070859148944145980986021, 15.04171876152247373200995289323, 15.518515893272895322320448665, 16.04989338124148610239827019621, 16.64306037700295037065819580886, 17.2391658663378775513709721259, 17.7959850875661026396638127373, 18.99701891094829576943494221010

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