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Properties

Label 1-2805-2805.59-r1-0-0
Degree $1$
Conductor $2805$
Sign $-0.363 - 0.931i$
Analytic cond. $301.439$
Root an. cond. $301.439$
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  + (0.951 − 0.309i)2-s + (0.809 − 0.587i)4-s + (0.156 − 0.987i)7-s + (0.587 − 0.809i)8-s + (0.309 + 0.951i)13-s + (−0.156 − 0.987i)14-s + (0.309 − 0.951i)16-s + (−0.587 + 0.809i)19-s + (−0.707 + 0.707i)23-s + (0.587 + 0.809i)26-s + (−0.453 − 0.891i)28-s + (0.987 + 0.156i)29-s + (0.453 − 0.891i)31-s − i·32-s + (−0.987 − 0.156i)37-s + (−0.309 + 0.951i)38-s + ⋯
L(s)  = 1  + (0.951 − 0.309i)2-s + (0.809 − 0.587i)4-s + (0.156 − 0.987i)7-s + (0.587 − 0.809i)8-s + (0.309 + 0.951i)13-s + (−0.156 − 0.987i)14-s + (0.309 − 0.951i)16-s + (−0.587 + 0.809i)19-s + (−0.707 + 0.707i)23-s + (0.587 + 0.809i)26-s + (−0.453 − 0.891i)28-s + (0.987 + 0.156i)29-s + (0.453 − 0.891i)31-s − i·32-s + (−0.987 − 0.156i)37-s + (−0.309 + 0.951i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2805 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.363 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2805 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.363 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2805\)    =    \(3 \cdot 5 \cdot 11 \cdot 17\)
Sign: $-0.363 - 0.931i$
Analytic conductor: \(301.439\)
Root analytic conductor: \(301.439\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2805} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2805,\ (1:\ ),\ -0.363 - 0.931i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.479616244 - 3.627205916i\)
\(L(\frac12)\) \(\approx\) \(2.479616244 - 3.627205916i\)
\(L(1)\) \(\approx\) \(1.821471863 - 0.7541239828i\)
\(L(1)\) \(\approx\) \(1.821471863 - 0.7541239828i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
11 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.951 - 0.309i)T \)
7 \( 1 + (0.156 - 0.987i)T \)
13 \( 1 + (0.309 + 0.951i)T \)
19 \( 1 + (-0.587 + 0.809i)T \)
23 \( 1 + (-0.707 + 0.707i)T \)
29 \( 1 + (0.987 + 0.156i)T \)
31 \( 1 + (0.453 - 0.891i)T \)
37 \( 1 + (-0.987 - 0.156i)T \)
41 \( 1 + (0.987 - 0.156i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.809 + 0.587i)T \)
53 \( 1 + (0.951 - 0.309i)T \)
59 \( 1 + (0.587 + 0.809i)T \)
61 \( 1 + (-0.453 - 0.891i)T \)
67 \( 1 - T \)
71 \( 1 + (0.891 - 0.453i)T \)
73 \( 1 + (0.987 + 0.156i)T \)
79 \( 1 + (-0.891 - 0.453i)T \)
83 \( 1 + (-0.951 - 0.309i)T \)
89 \( 1 + T \)
97 \( 1 + (0.453 - 0.891i)T \)
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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

−19.51207110680457798386405313163, −18.40763921948480052329823699860, −17.76075086539373350892433229348, −17.135220874673789110489814414362, −16.03794863540765311288124534998, −15.70730145802969560775280301512, −15.02502360396345649665101606310, −14.36389771748082199724095204122, −13.61259821910778137948973077197, −12.81526266037732369545347331829, −12.29753085508516374721631586973, −11.6482334328360945066975062870, −10.78782935743328508350733768864, −10.15176545250084296563441575109, −8.82002986479301087960976213758, −8.402868856407085217833558224441, −7.57116766809330323979271972542, −6.56664521684590301810442470807, −6.05947011655904286606966267872, −5.22183443558465525730294461008, −4.63536942053877455028387788592, −3.668079298700988167593774903289, −2.72563782872163260071216679846, −2.28714922050953113504588993501, −0.99686052081407352301515892154, 0.492363711547260403770611148082, 1.50765075991871916350825233285, 2.16729733162860088053723098298, 3.31884462681914949597690481856, 4.095823538864970186755051227717, 4.442168512043714749243790620059, 5.57198343090628814645647223057, 6.25624388225446105615469300992, 7.03092958541034512818335043658, 7.684273950909833441340552268109, 8.68696520644835643821880993149, 9.77393174493889569513372334726, 10.36567284426926936273225543172, 11.03283659732611031520059506826, 11.78650003242491874177021744810, 12.3913991559442293931432919603, 13.2698371111342380146732939576, 14.05697676614814492073662683507, 14.12932502847024991486659172246, 15.223724653833741194956585922557, 15.90307040143697058162062441011, 16.62028516496848758317328597823, 17.218428918187659178003991292449, 18.210305742689537731724681274139, 19.144793549027145559413692604409

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