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Properties

Label 1-6145-6145.1038-r0-0-0
Degree $1$
Conductor $6145$
Sign $0.961 - 0.274i$
Analytic cond. $28.5372$
Root an. cond. $28.5372$
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  + (−0.719 − 0.694i)2-s + (0.555 + 0.831i)3-s + (0.0358 + 0.999i)4-s + (0.178 − 0.984i)6-s + (−0.633 − 0.774i)7-s + (0.668 − 0.744i)8-s + (−0.383 + 0.923i)9-s + (−0.985 − 0.168i)11-s + (−0.811 + 0.584i)12-s + (0.637 + 0.770i)13-s + (−0.0817 + 0.996i)14-s + (−0.997 + 0.0715i)16-s + (−0.946 − 0.321i)17-s + (0.917 − 0.397i)18-s + (−0.550 − 0.834i)19-s + ⋯
L(s)  = 1  + (−0.719 − 0.694i)2-s + (0.555 + 0.831i)3-s + (0.0358 + 0.999i)4-s + (0.178 − 0.984i)6-s + (−0.633 − 0.774i)7-s + (0.668 − 0.744i)8-s + (−0.383 + 0.923i)9-s + (−0.985 − 0.168i)11-s + (−0.811 + 0.584i)12-s + (0.637 + 0.770i)13-s + (−0.0817 + 0.996i)14-s + (−0.997 + 0.0715i)16-s + (−0.946 − 0.321i)17-s + (0.917 − 0.397i)18-s + (−0.550 − 0.834i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.961 - 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.961 - 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6145\)    =    \(5 \cdot 1229\)
Sign: $0.961 - 0.274i$
Analytic conductor: \(28.5372\)
Root analytic conductor: \(28.5372\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6145} (1038, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6145,\ (0:\ ),\ 0.961 - 0.274i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7085196969 - 0.09905009484i\)
\(L(\frac12)\) \(\approx\) \(0.7085196969 - 0.09905009484i\)
\(L(1)\) \(\approx\) \(0.6655279796 + 0.02920810765i\)
\(L(1)\) \(\approx\) \(0.6655279796 + 0.02920810765i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
1229 \( 1 \)
good2 \( 1 + (-0.719 - 0.694i)T \)
3 \( 1 + (0.555 + 0.831i)T \)
7 \( 1 + (-0.633 - 0.774i)T \)
11 \( 1 + (-0.985 - 0.168i)T \)
13 \( 1 + (0.637 + 0.770i)T \)
17 \( 1 + (-0.946 - 0.321i)T \)
19 \( 1 + (-0.550 - 0.834i)T \)
23 \( 1 + (-0.953 - 0.302i)T \)
29 \( 1 + (0.625 - 0.780i)T \)
31 \( 1 + (0.430 + 0.902i)T \)
37 \( 1 + (-0.913 + 0.407i)T \)
41 \( 1 + (-0.998 + 0.0613i)T \)
43 \( 1 + (-0.884 - 0.467i)T \)
47 \( 1 + (-0.723 + 0.690i)T \)
53 \( 1 + (0.941 + 0.336i)T \)
59 \( 1 + (-0.345 + 0.938i)T \)
61 \( 1 + (0.520 - 0.853i)T \)
67 \( 1 + (0.656 - 0.754i)T \)
71 \( 1 + (0.820 + 0.571i)T \)
73 \( 1 + (0.814 + 0.580i)T \)
79 \( 1 + (-0.668 - 0.744i)T \)
83 \( 1 + (0.805 + 0.592i)T \)
89 \( 1 + (-0.969 - 0.243i)T \)
97 \( 1 + (0.183 + 0.983i)T \)
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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

−18.03107393763280605587622005318, −17.1949783038450164273557166804, −16.30167754519151681470089421366, −15.65909167184485624668025565970, −15.22967045456592813887640166832, −14.662188734976082142108006466402, −13.67653808560465370487355546018, −13.27447078205545867472612551310, −12.59564557968674838897274486854, −11.871235373787400876486987620862, −10.95741323032450650125510768456, −10.14929253504823869174116669000, −9.70219304270070638943347318159, −8.640126322361175854300422128683, −8.39358500287363378602061497353, −7.89224330601185704009925195270, −6.88469281243037413640880341239, −6.45494467876468386612863335270, −5.74553916542179232187670710863, −5.19186090390772900433276559602, −3.91309786025181710104888111204, −3.04917102598853405051165618950, −2.16533833650109825222811795814, −1.73510023137320641693929624210, −0.47325339880214418565931030620, 0.39474781513078358138155909035, 1.70117211767471577937547226404, 2.48494113481387282327410298805, 3.07330885708211643618023270138, 3.8694368464200745435824981113, 4.395342225562738691595306355463, 5.1144206404322179882495133279, 6.532721945208535458169295902198, 6.89387556313873165532219283866, 8.01316415116072181642526077302, 8.42099795917939850147811210003, 9.0707264758499108130434542009, 9.80242119251352219366766628436, 10.32968732319836544557772775045, 10.815904030506576874960770979233, 11.432806770106482891061746950436, 12.28551869439120449366336314855, 13.31686418896909168242889116703, 13.529500211659950137348147949897, 14.11112093854380389525266923569, 15.50175117849483929121946452475, 15.69043558612883405037242119421, 16.35101694613470008123890909043, 16.96586701987840915219760856672, 17.63655116779418025536348155808

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