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Properties

Label 1-4015-4015.2507-r0-0-0
Degree $1$
Conductor $4015$
Sign $-0.314 - 0.949i$
Analytic cond. $18.6455$
Root an. cond. $18.6455$
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  + (0.984 − 0.173i)2-s + (0.866 − 0.5i)3-s + (0.939 − 0.342i)4-s + (0.766 − 0.642i)6-s + (−0.5 − 0.866i)7-s + (0.866 − 0.5i)8-s + (0.5 − 0.866i)9-s + (0.642 − 0.766i)12-s + (0.766 − 0.642i)13-s + (−0.642 − 0.766i)14-s + (0.766 − 0.642i)16-s + (−0.5 + 0.866i)17-s + (0.342 − 0.939i)18-s + (−0.173 − 0.984i)19-s + (−0.866 − 0.5i)21-s + ⋯
L(s)  = 1  + (0.984 − 0.173i)2-s + (0.866 − 0.5i)3-s + (0.939 − 0.342i)4-s + (0.766 − 0.642i)6-s + (−0.5 − 0.866i)7-s + (0.866 − 0.5i)8-s + (0.5 − 0.866i)9-s + (0.642 − 0.766i)12-s + (0.766 − 0.642i)13-s + (−0.642 − 0.766i)14-s + (0.766 − 0.642i)16-s + (−0.5 + 0.866i)17-s + (0.342 − 0.939i)18-s + (−0.173 − 0.984i)19-s + (−0.866 − 0.5i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.314 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.314 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4015\)    =    \(5 \cdot 11 \cdot 73\)
Sign: $-0.314 - 0.949i$
Analytic conductor: \(18.6455\)
Root analytic conductor: \(18.6455\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4015} (2507, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4015,\ (0:\ ),\ -0.314 - 0.949i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.964836836 - 4.104314868i\)
\(L(\frac12)\) \(\approx\) \(2.964836836 - 4.104314868i\)
\(L(1)\) \(\approx\) \(2.303572930 - 1.292917054i\)
\(L(1)\) \(\approx\) \(2.303572930 - 1.292917054i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
73 \( 1 \)
good2 \( 1 + (0.984 - 0.173i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (0.766 - 0.642i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-0.173 - 0.984i)T \)
23 \( 1 + (0.984 + 0.173i)T \)
29 \( 1 + (0.984 - 0.173i)T \)
31 \( 1 + (-0.342 + 0.939i)T \)
37 \( 1 + (0.984 + 0.173i)T \)
41 \( 1 + (-0.766 - 0.642i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (0.766 + 0.642i)T \)
53 \( 1 + (0.173 + 0.984i)T \)
59 \( 1 + (-0.642 - 0.766i)T \)
61 \( 1 + (0.766 + 0.642i)T \)
67 \( 1 + (-0.642 - 0.766i)T \)
71 \( 1 + (0.173 + 0.984i)T \)
79 \( 1 + (-0.766 + 0.642i)T \)
83 \( 1 - T \)
89 \( 1 + (-0.766 + 0.642i)T \)
97 \( 1 + (0.866 + 0.5i)T \)
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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

−18.734151554284771765577951056317, −18.278594452078786666118208161965, −16.88862094563330451710310270811, −16.32343271519889749683357135348, −15.86068815166351024773406854353, −15.15528778326432499971519182529, −14.65420592783128709925989080918, −13.97327598963583558082520361036, −13.21017762654573161979157453292, −12.87839273562584250359140840393, −11.80359757078364944857694228242, −11.35142444886451578557629176612, −10.40652762384982244480135812230, −9.65197224532873969624609825968, −8.84859726812483088665196584269, −8.3173123890106784351248560764, −7.40842581403366742730658012899, −6.60817935186438011707332846312, −5.957395496321820929576573794439, −5.05730411180333044985915632277, −4.413669449996391008034272729391, −3.63274695738457423788548742396, −2.94013078032886436526522923004, −2.34529341856122808492871262443, −1.478424882548821333321493882763, 0.85769172663927537318160436920, 1.5088073175864520111854040506, 2.62928533757102854660520554135, 3.134032649533832466442112908166, 3.89357591943231003780677825035, 4.48037158413545253318846485780, 5.54026695402680573800057654430, 6.50599520720284630558151274496, 6.84282891841878983185100326673, 7.60369044537338927331945719249, 8.454312264185983520259375417862, 9.183085406457187736496739028942, 10.239442522612505198967118306508, 10.67676759382818290092020481997, 11.46968120812689418286410296585, 12.53326067967308122226935172300, 12.90121702601597330326257256499, 13.53418014219257682174280729000, 13.92133028307583939911289144722, 14.76839606420505069298788711819, 15.52150244974671314142694473468, 15.76668326321150143433691108233, 16.93412257524882495770456814579, 17.50362220568027431614884509561, 18.50958223557896629583208043667

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