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Properties

Label 1-1035-1035.718-r0-0-0
Degree $1$
Conductor $1035$
Sign $0.0442 + 0.999i$
Analytic cond. $4.80651$
Root an. cond. $4.80651$
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  + (−0.998 − 0.0475i)2-s + (0.995 + 0.0950i)4-s + (−0.189 + 0.981i)7-s + (−0.989 − 0.142i)8-s + (0.888 + 0.458i)11-s + (0.189 + 0.981i)13-s + (0.235 − 0.971i)14-s + (0.981 + 0.189i)16-s + (0.909 + 0.415i)17-s + (0.415 + 0.909i)19-s + (−0.866 − 0.5i)22-s + (−0.142 − 0.989i)26-s + (−0.281 + 0.959i)28-s + (0.995 − 0.0950i)29-s + (−0.786 − 0.618i)31-s + (−0.971 − 0.235i)32-s + ⋯
L(s)  = 1  + (−0.998 − 0.0475i)2-s + (0.995 + 0.0950i)4-s + (−0.189 + 0.981i)7-s + (−0.989 − 0.142i)8-s + (0.888 + 0.458i)11-s + (0.189 + 0.981i)13-s + (0.235 − 0.971i)14-s + (0.981 + 0.189i)16-s + (0.909 + 0.415i)17-s + (0.415 + 0.909i)19-s + (−0.866 − 0.5i)22-s + (−0.142 − 0.989i)26-s + (−0.281 + 0.959i)28-s + (0.995 − 0.0950i)29-s + (−0.786 − 0.618i)31-s + (−0.971 − 0.235i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0442 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0442 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1035\)    =    \(3^{2} \cdot 5 \cdot 23\)
Sign: $0.0442 + 0.999i$
Analytic conductor: \(4.80651\)
Root analytic conductor: \(4.80651\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1035} (718, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1035,\ (0:\ ),\ 0.0442 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7039015056 + 0.6733926332i\)
\(L(\frac12)\) \(\approx\) \(0.7039015056 + 0.6733926332i\)
\(L(1)\) \(\approx\) \(0.7368045238 + 0.2151034959i\)
\(L(1)\) \(\approx\) \(0.7368045238 + 0.2151034959i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.998 - 0.0475i)T \)
7 \( 1 + (-0.189 + 0.981i)T \)
11 \( 1 + (0.888 + 0.458i)T \)
13 \( 1 + (0.189 + 0.981i)T \)
17 \( 1 + (0.909 + 0.415i)T \)
19 \( 1 + (0.415 + 0.909i)T \)
29 \( 1 + (0.995 - 0.0950i)T \)
31 \( 1 + (-0.786 - 0.618i)T \)
37 \( 1 + (-0.281 - 0.959i)T \)
41 \( 1 + (0.723 - 0.690i)T \)
43 \( 1 + (0.618 + 0.786i)T \)
47 \( 1 + (-0.866 + 0.5i)T \)
53 \( 1 + (0.755 + 0.654i)T \)
59 \( 1 + (-0.981 + 0.189i)T \)
61 \( 1 + (-0.928 + 0.371i)T \)
67 \( 1 + (-0.458 - 0.888i)T \)
71 \( 1 + (0.841 + 0.540i)T \)
73 \( 1 + (-0.909 + 0.415i)T \)
79 \( 1 + (-0.327 + 0.945i)T \)
83 \( 1 + (0.690 - 0.723i)T \)
89 \( 1 + (-0.142 - 0.989i)T \)
97 \( 1 + (-0.971 + 0.235i)T \)
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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

−21.15105542672624830430993740507, −20.29367286669988220501133454548, −19.788995067444170767747819542630, −19.151360268193464777678560637835, −18.09369926374510089471180041840, −17.55569021990493680503629552644, −16.69224360701881150735080946717, −16.204145288593196290441673081268, −15.27328188483662693190615747892, −14.34744265809752829367426719096, −13.56237817226038110129342000282, −12.465773534862081583605315520022, −11.608114898066932093992629405587, −10.80061634057417763219022891870, −10.125623368038305350595748955329, −9.33547560328638770112017913853, −8.45351674876243395941853116662, −7.59590158311005921296677838854, −6.89258073742816645308544676083, −6.058622518829147572128185226059, −4.95693558209648761422797216241, −3.53395337181989825074641396970, −2.93346091890176894331049283254, −1.37519085751962065423712920668, −0.6583998551433197434412682374, 1.301326125232047670198109227541, 2.05480853917420238595533196284, 3.16159629324266887946676504615, 4.18850347472502942832569978796, 5.70056016603982620848556717331, 6.25533828343295840474534740391, 7.25580709998025966145435296009, 8.0924719892925276554034429317, 9.10061413337604386771480470756, 9.44211472699767892558913960806, 10.4052790633086520195663468720, 11.42805900304490970600652828655, 12.11713709908439278385325162680, 12.59400643796532661570843186271, 14.19784341743851151973820927712, 14.72736378593103153296027851022, 15.71975783128441718189853973416, 16.39816993025103453368871028036, 17.06091171787943309673408405745, 18.00152675700166334027996588557, 18.64428439054501758006495002586, 19.31215660812779420597506539417, 19.927720088937878401558322834018, 21.10825038234393253771391063266, 21.377664381042479843026788674197

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