Properties

Label 2-1350-45.23-c1-0-11
Degree $2$
Conductor $1350$
Sign $0.292 + 0.956i$
Analytic cond. $10.7798$
Root an. cond. $3.28326$
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  + (0.258 − 0.965i)2-s + (−0.866 − 0.499i)4-s + (1.05 + 0.283i)7-s + (−0.707 + 0.707i)8-s + (5.44 − 3.14i)11-s + (−3.34 + 0.896i)13-s + (0.548 − 0.949i)14-s + (0.500 + 0.866i)16-s + (3.14 + 3.14i)17-s − 1.55i·19-s + (−1.62 − 6.07i)22-s + (0.258 + 0.965i)23-s + 3.46i·26-s + (−0.775 − 0.775i)28-s + (1.57 + 2.72i)29-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (−0.433 − 0.249i)4-s + (0.400 + 0.107i)7-s + (−0.249 + 0.249i)8-s + (1.64 − 0.948i)11-s + (−0.928 + 0.248i)13-s + (0.146 − 0.253i)14-s + (0.125 + 0.216i)16-s + (0.763 + 0.763i)17-s − 0.355i·19-s + (−0.347 − 1.29i)22-s + (0.0539 + 0.201i)23-s + 0.679i·26-s + (−0.146 − 0.146i)28-s + (0.292 + 0.505i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.292 + 0.956i$
Analytic conductor: \(10.7798\)
Root analytic conductor: \(3.28326\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1350} (1043, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1350,\ (\ :1/2),\ 0.292 + 0.956i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.950781383\)
\(L(\frac12)\) \(\approx\) \(1.950781383\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 + 0.965i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-1.05 - 0.283i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-5.44 + 3.14i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.34 - 0.896i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (-3.14 - 3.14i)T + 17iT^{2} \)
19 \( 1 + 1.55iT - 19T^{2} \)
23 \( 1 + (-0.258 - 0.965i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-1.57 - 2.72i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.22 + 3.85i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3 + 3i)T - 37iT^{2} \)
41 \( 1 + (-3.39 - 1.96i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.896 - 3.34i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-2.32 + 8.69i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-6.61 + 6.61i)T - 53iT^{2} \)
59 \( 1 + (-5.90 + 10.2i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.72 - 4.71i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.978 + 3.65i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 0.635iT - 71T^{2} \)
73 \( 1 + (2.89 + 2.89i)T + 73iT^{2} \)
79 \( 1 + (-2.12 + 1.22i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.531 + 0.142i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 - 2.36T + 89T^{2} \)
97 \( 1 + (10.7 + 2.89i)T + (84.0 + 48.5i)T^{2} \)
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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

−9.514773185602843741136045435445, −8.778673375351725453440790964827, −8.028384271259631787654811355105, −6.89222627604337485964230894425, −6.04428473217149151617943889480, −5.12500914232682303358879676419, −4.11554903552083719330370734356, −3.36973176424087594016923806858, −2.10571581220608851713952195022, −0.948331777322512964515773527253, 1.21401811998272364604304550108, 2.72236579935696915734426173406, 4.05036087087519238018151395111, 4.67549182316166497995506849744, 5.62422228654854926874862254729, 6.62079662668333269874977335327, 7.27386762319272082954577141653, 7.953133110734823246969546057027, 8.986739471365720376635996285592, 9.632090574785397567852264844089

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