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Properties

Label 2-2700-15.14-c0-0-2
Degree $2$
Conductor $2700$
Sign $0.894 + 0.447i$
Analytic cond. $1.34747$
Root an. cond. $1.16080$
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  − i·7-s + i·13-s + 19-s + 2·31-s − i·37-s − 2i·43-s − 61-s − i·67-s + i·73-s + 79-s + 91-s − i·97-s + i·103-s − 2·109-s + ⋯
L(s)  = 1  − i·7-s + i·13-s + 19-s + 2·31-s − i·37-s − 2i·43-s − 61-s − i·67-s + i·73-s + 79-s + 91-s − i·97-s + i·103-s − 2·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2700\)    =    \(2^{2} \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(1.34747\)
Root analytic conductor: \(1.16080\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2700} (1349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2700,\ (\ :0),\ 0.894 + 0.447i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.255282400\)
\(L(\frac12)\) \(\approx\) \(1.255282400\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + iT - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - iT - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - 2T + T^{2} \)
37 \( 1 + iT - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + 2iT - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 + iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - iT - T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + iT - T^{2} \)
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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

−9.022968386281442790629243473510, −8.159352355633623469613149979578, −7.34268779113398406853526698559, −6.84110388173245529490790083993, −5.95778874300226311195795618632, −4.94869727530295963056472943606, −4.19521237190562697050422911881, −3.42921965006363325761873268863, −2.23806043496368721257495663559, −0.994095892844798570343140418950, 1.22909762722202828653154492068, 2.66251249328726856360676419346, 3.15287480731461625651359858298, 4.48519790063382472256866048148, 5.24512422147735238970757732498, 5.97359831197246788543271792765, 6.67339058028697964229624316996, 7.83042033508815096795660670505, 8.182763588935625742386461786840, 9.106395684223277870978637308001

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