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Properties

Label 2-3087-49.6-c0-0-0
Degree $2$
Conductor $3087$
Sign $-0.0960 - 0.995i$
Analytic cond. $1.54061$
Root an. cond. $1.24121$
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  + (0.222 + 0.974i)4-s + (−0.678 − 0.541i)13-s + (−0.900 + 0.433i)16-s + 1.56i·19-s + (0.623 + 0.781i)25-s + 1.94i·31-s + (−0.0990 + 0.433i)37-s + (1.12 − 0.541i)43-s + (0.376 − 0.781i)52-s + (1.90 + 0.433i)61-s + (−0.623 − 0.781i)64-s − 1.80·67-s + (−1.52 + 1.21i)73-s + (−1.52 + 0.347i)76-s − 0.445·79-s + ⋯
L(s)  = 1  + (0.222 + 0.974i)4-s + (−0.678 − 0.541i)13-s + (−0.900 + 0.433i)16-s + 1.56i·19-s + (0.623 + 0.781i)25-s + 1.94i·31-s + (−0.0990 + 0.433i)37-s + (1.12 − 0.541i)43-s + (0.376 − 0.781i)52-s + (1.90 + 0.433i)61-s + (−0.623 − 0.781i)64-s − 1.80·67-s + (−1.52 + 1.21i)73-s + (−1.52 + 0.347i)76-s − 0.445·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0960 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0960 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3087\)    =    \(3^{2} \cdot 7^{3}\)
Sign: $-0.0960 - 0.995i$
Analytic conductor: \(1.54061\)
Root analytic conductor: \(1.24121\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3087} (2449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3087,\ (\ :0),\ -0.0960 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.133274232\)
\(L(\frac12)\) \(\approx\) \(1.133274232\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.222 - 0.974i)T^{2} \)
5 \( 1 + (-0.623 - 0.781i)T^{2} \)
11 \( 1 + (-0.222 - 0.974i)T^{2} \)
13 \( 1 + (0.678 + 0.541i)T + (0.222 + 0.974i)T^{2} \)
17 \( 1 + (0.900 + 0.433i)T^{2} \)
19 \( 1 - 1.56iT - T^{2} \)
23 \( 1 + (-0.900 + 0.433i)T^{2} \)
29 \( 1 + (-0.900 - 0.433i)T^{2} \)
31 \( 1 - 1.94iT - T^{2} \)
37 \( 1 + (0.0990 - 0.433i)T + (-0.900 - 0.433i)T^{2} \)
41 \( 1 + (-0.623 - 0.781i)T^{2} \)
43 \( 1 + (-1.12 + 0.541i)T + (0.623 - 0.781i)T^{2} \)
47 \( 1 + (0.222 + 0.974i)T^{2} \)
53 \( 1 + (-0.900 + 0.433i)T^{2} \)
59 \( 1 + (-0.623 + 0.781i)T^{2} \)
61 \( 1 + (-1.90 - 0.433i)T + (0.900 + 0.433i)T^{2} \)
67 \( 1 + 1.80T + T^{2} \)
71 \( 1 + (-0.900 + 0.433i)T^{2} \)
73 \( 1 + (1.52 - 1.21i)T + (0.222 - 0.974i)T^{2} \)
79 \( 1 + 0.445T + T^{2} \)
83 \( 1 + (0.222 - 0.974i)T^{2} \)
89 \( 1 + (0.222 - 0.974i)T^{2} \)
97 \( 1 + 1.56iT - T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.751428253769570714847926524507, −8.461820911684394162112431852805, −7.39437952148996482484997031862, −7.18132147662143684383078607656, −6.07133808857271238721972951919, −5.25289076291941425437984163181, −4.31860051209683860729003095577, −3.42447369998823907513511783966, −2.77620438531652123667595692004, −1.57521803955966105697309942107, 0.68966068024091917362560834914, 2.10349709561936490900434023998, 2.74741729824264630179591315228, 4.24594447658004309516263897647, 4.79298121973708606324202419459, 5.68165662523468793274248552554, 6.42770906292243772436791629723, 7.08175000630114279148305912324, 7.81355570207801181148255728208, 9.008767691097463758747837579368

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