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Properties

Label 2-2700-45.34-c1-0-17
Degree $2$
Conductor $2700$
Sign $-0.947 - 0.319i$
Analytic cond. $21.5596$
Root an. cond. $4.64323$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.32 − 2.49i)7-s + (1.99 − 3.46i)11-s + (1.33 − 0.771i)13-s − 6.99i·17-s + 2.25·19-s + (−6.75 + 3.89i)23-s + (−3.08 + 5.33i)29-s + (0.271 + 0.470i)31-s + 6.25i·37-s + (0.0979 + 0.169i)41-s + (0.0747 + 0.0431i)43-s + (3.31 + 1.91i)47-s + (8.97 + 15.5i)49-s − 4.19i·53-s + (−3.51 − 6.08i)59-s + ⋯
L(s)  = 1  + (−1.63 − 0.944i)7-s + (0.602 − 1.04i)11-s + (0.370 − 0.213i)13-s − 1.69i·17-s + 0.517·19-s + (−1.40 + 0.812i)23-s + (−0.572 + 0.991i)29-s + (0.0487 + 0.0844i)31-s + 1.02i·37-s + (0.0152 + 0.0264i)41-s + (0.0114 + 0.00658i)43-s + (0.483 + 0.279i)47-s + (1.28 + 2.22i)49-s − 0.575i·53-s + (−0.457 − 0.792i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.947 - 0.319i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.947 - 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2700\)    =    \(2^{2} \cdot 3^{3} \cdot 5^{2}\)
Sign: $-0.947 - 0.319i$
Analytic conductor: \(21.5596\)
Root analytic conductor: \(4.64323\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2700} (2449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2700,\ (\ :1/2),\ -0.947 - 0.319i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3358013735\)
\(L(\frac12)\) \(\approx\) \(0.3358013735\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.32 + 2.49i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.99 + 3.46i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.33 + 0.771i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 6.99iT - 17T^{2} \)
19 \( 1 - 2.25T + 19T^{2} \)
23 \( 1 + (6.75 - 3.89i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.08 - 5.33i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.271 - 0.470i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 6.25iT - 37T^{2} \)
41 \( 1 + (-0.0979 - 0.169i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.0747 - 0.0431i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.31 - 1.91i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 4.19iT - 53T^{2} \)
59 \( 1 + (3.51 + 6.08i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.45 + 2.51i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.76 - 4.48i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.79T + 71T^{2} \)
73 \( 1 + 2.28iT - 73T^{2} \)
79 \( 1 + (6.32 - 10.9i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (12.0 + 6.98i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 10.3T + 89T^{2} \)
97 \( 1 + (-8.08 - 4.66i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.490830253916914808445557584746, −7.46444331366951202669469575759, −6.93590065059463040744328002965, −6.16023260883458253962837588302, −5.49194999438113699472119916622, −4.25242131345042873716716732070, −3.39959967851210565589634786332, −2.98567251659223811530972950148, −1.20718481036976563786621652922, −0.11436202622165541586648467811, 1.76282237805173923491457584832, 2.63792312100526906869333519671, 3.77861815633665954259215572309, 4.26462617272913806741523844279, 5.85831298943356097383009936158, 5.99779047354936838117565894982, 6.84187849898938889475751820101, 7.71735350484178334735598690175, 8.722551299949830724909404046207, 9.194646412981323381438516990616

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