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Properties

Label 1-6037-6037.6-r1-0-0
Degree $1$
Conductor $6037$
Sign $0.922 + 0.386i$
Analytic cond. $648.765$
Root an. cond. $648.765$
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  + (0.235 − 0.971i)2-s + (0.510 + 0.859i)3-s + (−0.889 − 0.457i)4-s + (−0.249 + 0.968i)5-s + (0.955 − 0.294i)6-s + (0.893 − 0.448i)7-s + (−0.653 + 0.757i)8-s + (−0.478 + 0.878i)9-s + (0.882 + 0.469i)10-s + (−0.631 + 0.775i)11-s + (−0.0613 − 0.998i)12-s + (0.858 + 0.512i)13-s + (−0.225 − 0.974i)14-s + (−0.959 + 0.280i)15-s + (0.582 + 0.813i)16-s + (0.888 − 0.458i)17-s + ⋯
L(s)  = 1  + (0.235 − 0.971i)2-s + (0.510 + 0.859i)3-s + (−0.889 − 0.457i)4-s + (−0.249 + 0.968i)5-s + (0.955 − 0.294i)6-s + (0.893 − 0.448i)7-s + (−0.653 + 0.757i)8-s + (−0.478 + 0.878i)9-s + (0.882 + 0.469i)10-s + (−0.631 + 0.775i)11-s + (−0.0613 − 0.998i)12-s + (0.858 + 0.512i)13-s + (−0.225 − 0.974i)14-s + (−0.959 + 0.280i)15-s + (0.582 + 0.813i)16-s + (0.888 − 0.458i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6037 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.922 + 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6037 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.922 + 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6037\)
Sign: $0.922 + 0.386i$
Analytic conductor: \(648.765\)
Root analytic conductor: \(648.765\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6037} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6037,\ (1:\ ),\ 0.922 + 0.386i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.663958118 + 0.5353049008i\)
\(L(\frac12)\) \(\approx\) \(2.663958118 + 0.5353049008i\)
\(L(1)\) \(\approx\) \(1.281076086 - 0.03507740304i\)
\(L(1)\) \(\approx\) \(1.281076086 - 0.03507740304i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad6037 \( 1 \)
good2 \( 1 + (0.235 - 0.971i)T \)
3 \( 1 + (0.510 + 0.859i)T \)
5 \( 1 + (-0.249 + 0.968i)T \)
7 \( 1 + (0.893 - 0.448i)T \)
11 \( 1 + (-0.631 + 0.775i)T \)
13 \( 1 + (0.858 + 0.512i)T \)
17 \( 1 + (0.888 - 0.458i)T \)
19 \( 1 + (-0.746 - 0.665i)T \)
23 \( 1 + (-0.477 + 0.878i)T \)
29 \( 1 + (-0.944 - 0.328i)T \)
31 \( 1 + (0.442 - 0.896i)T \)
37 \( 1 + (-0.391 - 0.920i)T \)
41 \( 1 + (-0.926 - 0.375i)T \)
43 \( 1 + (-0.672 - 0.739i)T \)
47 \( 1 + (0.887 + 0.460i)T \)
53 \( 1 + (-0.176 - 0.984i)T \)
59 \( 1 + (0.0676 + 0.997i)T \)
61 \( 1 + (0.810 + 0.585i)T \)
67 \( 1 + (0.993 + 0.117i)T \)
71 \( 1 + (0.895 + 0.445i)T \)
73 \( 1 + (0.0644 + 0.997i)T \)
79 \( 1 + (0.493 - 0.869i)T \)
83 \( 1 + (0.833 - 0.552i)T \)
89 \( 1 + (0.960 - 0.278i)T \)
97 \( 1 + (0.196 + 0.980i)T \)
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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

−17.39201060182490775375536206932, −16.81842457245750960116694499071, −16.24274662954922625137394621106, −15.34260722678253383337397903379, −15.01778729870753576610111445029, −14.09281359455480735622762678995, −13.748082747921077398834014165061, −12.89922063614842504249617119208, −12.51021231652757854481996061418, −11.945721783756030801378965547275, −11.00665630317470599122837821078, −10.02369916086087354590437561210, −9.00306996090618481343353019613, −8.34204882971687516702252194341, −8.22650045889038993202771232687, −7.77649455591673050515008827787, −6.629753195290598218337253510396, −5.976236891137026163175038896894, −5.41292129523624346942010439917, −4.76528052655874165361272730486, −3.70913573484962790103076856893, −3.2619662570010017149123817741, −2.02496865046477489255567967062, −1.20788822828043024354540248809, −0.494571530472806997270441703, 0.56650345147462195780971445822, 1.98940139979991605253579863287, 2.14890043247595593037502963367, 3.19371502200468865634768035911, 3.90618250671264066168550992769, 4.20484886579241745922611795189, 5.17808980367795059591329078683, 5.67985669264656390527481195302, 6.93562601397609971261412572217, 7.73933411515849780372173247982, 8.275110601392572489951725625790, 9.14733201149574980773993804029, 9.84724440646715646142217774325, 10.393627361262339149790942451556, 10.922925272953207581676789686704, 11.474786823339958909373511061515, 12.00666102495582450425033616516, 13.29127490646861957325007388954, 13.61748180396505856984783258488, 14.322111509906920933591197621585, 14.86633990407115589073923335906, 15.33721185249553651622242671500, 16.061814199722637509481605272143, 17.18934365337939369227967461348, 17.61300688612284904573472587862

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