↑

Properties

Label 1-4235-4235.598-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.277 - 0.960i$
Analytic cond. $19.6672$
Root an. cond. $19.6672$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.603 − 0.797i)2-s + (0.994 + 0.104i)3-s + (−0.272 + 0.962i)4-s + (−0.516 − 0.856i)6-s + (0.931 − 0.362i)8-s + (0.978 + 0.207i)9-s + (−0.371 + 0.928i)12-s + (−0.441 − 0.897i)13-s + (−0.851 − 0.524i)16-s + (−0.976 − 0.217i)17-s + (−0.424 − 0.905i)18-s + (0.749 + 0.662i)19-s + (0.971 + 0.235i)23-s + (0.964 − 0.263i)24-s + (−0.449 + 0.893i)26-s + (0.951 + 0.309i)27-s + ⋯
L(s)  = 1  + (−0.603 − 0.797i)2-s + (0.994 + 0.104i)3-s + (−0.272 + 0.962i)4-s + (−0.516 − 0.856i)6-s + (0.931 − 0.362i)8-s + (0.978 + 0.207i)9-s + (−0.371 + 0.928i)12-s + (−0.441 − 0.897i)13-s + (−0.851 − 0.524i)16-s + (−0.976 − 0.217i)17-s + (−0.424 − 0.905i)18-s + (0.749 + 0.662i)19-s + (0.971 + 0.235i)23-s + (0.964 − 0.263i)24-s + (−0.449 + 0.893i)26-s + (0.951 + 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.277 - 0.960i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (598, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.277 - 0.960i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.436094880 - 1.080078653i\)
\(L(\frac12)\) \(\approx\) \(1.436094880 - 1.080078653i\)
\(L(1)\) \(\approx\) \(1.051134745 - 0.3791240843i\)
\(L(1)\) \(\approx\) \(1.051134745 - 0.3791240843i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.603 - 0.797i)T \)
3 \( 1 + (0.994 + 0.104i)T \)
13 \( 1 + (-0.441 - 0.897i)T \)
17 \( 1 + (-0.976 - 0.217i)T \)
19 \( 1 + (0.749 + 0.662i)T \)
23 \( 1 + (0.971 + 0.235i)T \)
29 \( 1 + (0.870 + 0.491i)T \)
31 \( 1 + (-0.640 - 0.768i)T \)
37 \( 1 + (0.0380 - 0.999i)T \)
41 \( 1 + (-0.974 + 0.226i)T \)
43 \( 1 + (-0.540 + 0.841i)T \)
47 \( 1 + (0.424 - 0.905i)T \)
53 \( 1 + (-0.524 - 0.851i)T \)
59 \( 1 + (-0.290 - 0.956i)T \)
61 \( 1 + (-0.123 + 0.992i)T \)
67 \( 1 + (0.189 - 0.981i)T \)
71 \( 1 + (0.993 - 0.113i)T \)
73 \( 1 + (-0.132 + 0.991i)T \)
79 \( 1 + (0.935 - 0.353i)T \)
83 \( 1 + (0.825 + 0.564i)T \)
89 \( 1 + (0.580 + 0.814i)T \)
97 \( 1 + (-0.967 - 0.254i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.6902189332329709940468230873, −17.787197284493162728390972209334, −17.21194672641510002706168916795, −16.432342788292097500386677085466, −15.64277916073829568186381517386, −15.27219587121160758306368618573, −14.548315052594803062289758012159, −13.79696315886602777010307296624, −13.51775868472645529363808428246, −12.53522609375354599564343732373, −11.60130305428849193566933974473, −10.71919938952339684742374426394, −10.0331587694393565714426037892, −9.17249643691416241043116732959, −8.94492514582158555649067143263, −8.175778621577100169987766772164, −7.331885893972597725514659953960, −6.86543639939515726106281576765, −6.25743331101775196102974053115, −4.895388008482172373159565139951, −4.655004984551150391831603242834, −3.52256732961359788727166893028, −2.54775698741910264476352497916, −1.798926417141267102567709773655, −0.91506646105011674060040756378, 0.633155574796526773918473237723, 1.65709133624125054922743205429, 2.35689792610754094383661849119, 3.15518584846903904538353741021, 3.61742631178186966558558813632, 4.60783153160237595516457668730, 5.27215601745768792109870916220, 6.70516437624330411545506857556, 7.35294458073441241663559286371, 8.0449838157387881487218214932, 8.58928622216948874229766844389, 9.41179675528232299382333820584, 9.80215357606694760457501689664, 10.61982907182668428165868454247, 11.206422365354074801288756225978, 12.16839255439466176362264558785, 12.79089477522855185428027706590, 13.367824960283646734386391910351, 14.006435960739367523732933107791, 14.87929711829135023299779389445, 15.50624484276131512863295840130, 16.27666005215303563658673832947, 16.94011798133877260819686177293, 17.96562444135908561925396202451, 18.16742318196844272938656898633

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