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Properties

Label 1-5735-5735.1164-r0-0-0
Degree $1$
Conductor $5735$
Sign $-0.894 - 0.446i$
Analytic cond. $26.6332$
Root an. cond. $26.6332$
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  + (−0.275 + 0.961i)2-s + (0.241 − 0.970i)3-s + (−0.848 − 0.529i)4-s + (0.866 + 0.5i)6-s + (−0.0348 + 0.999i)7-s + (0.743 − 0.669i)8-s + (−0.882 − 0.469i)9-s + (0.309 + 0.951i)11-s + (−0.719 + 0.694i)12-s + (0.275 + 0.961i)13-s + (−0.951 − 0.309i)14-s + (0.438 + 0.898i)16-s + (−0.999 + 0.0348i)17-s + (0.694 − 0.719i)18-s + (−0.0697 − 0.997i)19-s + ⋯
L(s)  = 1  + (−0.275 + 0.961i)2-s + (0.241 − 0.970i)3-s + (−0.848 − 0.529i)4-s + (0.866 + 0.5i)6-s + (−0.0348 + 0.999i)7-s + (0.743 − 0.669i)8-s + (−0.882 − 0.469i)9-s + (0.309 + 0.951i)11-s + (−0.719 + 0.694i)12-s + (0.275 + 0.961i)13-s + (−0.951 − 0.309i)14-s + (0.438 + 0.898i)16-s + (−0.999 + 0.0348i)17-s + (0.694 − 0.719i)18-s + (−0.0697 − 0.997i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5735 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5735 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5735\)    =    \(5 \cdot 31 \cdot 37\)
Sign: $-0.894 - 0.446i$
Analytic conductor: \(26.6332\)
Root analytic conductor: \(26.6332\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5735} (1164, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5735,\ (0:\ ),\ -0.894 - 0.446i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.06591252528 + 0.2798725471i\)
\(L(\frac12)\) \(\approx\) \(-0.06591252528 + 0.2798725471i\)
\(L(1)\) \(\approx\) \(0.7243390337 + 0.2443546362i\)
\(L(1)\) \(\approx\) \(0.7243390337 + 0.2443546362i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
31 \( 1 \)
37 \( 1 \)
good2 \( 1 + (-0.275 + 0.961i)T \)
3 \( 1 + (0.241 - 0.970i)T \)
7 \( 1 + (-0.0348 + 0.999i)T \)
11 \( 1 + (0.309 + 0.951i)T \)
13 \( 1 + (0.275 + 0.961i)T \)
17 \( 1 + (-0.999 + 0.0348i)T \)
19 \( 1 + (-0.0697 - 0.997i)T \)
23 \( 1 + (-0.207 - 0.978i)T \)
29 \( 1 + (0.406 + 0.913i)T \)
41 \( 1 + (-0.559 - 0.829i)T \)
43 \( 1 + (0.406 + 0.913i)T \)
47 \( 1 + (-0.913 - 0.406i)T \)
53 \( 1 + (0.0348 + 0.999i)T \)
59 \( 1 + (0.0697 - 0.997i)T \)
61 \( 1 + (0.642 + 0.766i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (0.990 + 0.139i)T \)
73 \( 1 + (-0.669 + 0.743i)T \)
79 \( 1 + (0.788 + 0.615i)T \)
83 \( 1 + (-0.719 + 0.694i)T \)
89 \( 1 + (0.139 + 0.990i)T \)
97 \( 1 + (-0.743 - 0.669i)T \)
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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

−17.40583368821892871629451822093, −16.884279621972014776007024603543, −16.18760446874898643474538923213, −15.586773862969528514056433826790, −14.61611395192343262709223889726, −14.018457201288452053163424813400, −13.41439246635551230383733564291, −12.97807587884293901843120728092, −11.72534708472797446467549425692, −11.37973898315103330215736635439, −10.61235822258225791694494123297, −10.22067510661076134061714655867, −9.59909747325466142307912439873, −8.82912306439576624705439181707, −8.1430223386521234961695845220, −7.74937341789690321953049211073, −6.47010619666297728233248952724, −5.60271729090296486053608699605, −4.844618321212445869441146146491, −4.00370409049708426680711008747, −3.608577034339397313894760814071, −3.01115540699132107137759817984, −2.03182448364994097203481271461, −1.0381025062437126760763828821, −0.08769877429178712007985983197, 1.203484402233913670244621435510, 2.019290994663306946752371924878, 2.6121397420937802851757266698, 3.86769525893489918782167035540, 4.70930768023124735710883709469, 5.32176145523273106039615494682, 6.43752210643251033286764102741, 6.588695474979700419678353031170, 7.19236916687672120937429292196, 8.15583085180767618908299945865, 8.76340497238771375453658626388, 9.1043777044144495591211503123, 9.83125140689164580417917431742, 10.97296682792280039371326166950, 11.62141811702468919132316082250, 12.58190615294881052065215770363, 12.79723569645507875892761171440, 13.78055976451015419880004111907, 14.24890996871025430830408775998, 14.9582956412681150790250814843, 15.45561189036228560245545481814, 16.17664205790123216766321732551, 16.986297455995464582431114028193, 17.6336275921497661666455711860, 18.16732014155730442498740937247

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