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Properties

Label 2-1350-45.2-c1-0-8
Degree $2$
Conductor $1350$
Sign $0.804 + 0.594i$
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 + (4.40 − 1.18i)7-s + (0.707 + 0.707i)8-s + (0.550 + 0.317i)11-s + (3.34 + 0.896i)13-s + (−2.28 − 3.94i)14-s + (0.500 − 0.866i)16-s + (0.317 − 0.317i)17-s + 6.44i·19-s + (0.164 − 0.614i)22-s + (−0.258 + 0.965i)23-s − 3.46i·26-s + (−3.22 + 3.22i)28-s + (0.158 − 0.275i)29-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (−0.433 + 0.249i)4-s + (1.66 − 0.446i)7-s + (0.249 + 0.249i)8-s + (0.165 + 0.0958i)11-s + (0.928 + 0.248i)13-s + (−0.609 − 1.05i)14-s + (0.125 − 0.216i)16-s + (0.0770 − 0.0770i)17-s + 1.47i·19-s + (0.0350 − 0.130i)22-s + (−0.0539 + 0.201i)23-s − 0.679i·26-s + (−0.609 + 0.609i)28-s + (0.0295 − 0.0511i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.804 + 0.594i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.804 + 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.804 + 0.594i$
Analytic conductor: \(10.7798\)
Root analytic conductor: \(3.28326\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1350} (1007, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1350,\ (\ :1/2),\ 0.804 + 0.594i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.862658166\)
\(L(\frac12)\) \(\approx\) \(1.862658166\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.40 + 1.18i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-0.550 - 0.317i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.34 - 0.896i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (-0.317 + 0.317i)T - 17iT^{2} \)
19 \( 1 - 6.44iT - 19T^{2} \)
23 \( 1 + (0.258 - 0.965i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-0.158 + 0.275i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.224 + 0.389i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3 - 3i)T + 37iT^{2} \)
41 \( 1 + (6.39 - 3.69i)T + (20.5 - 35.5i)T^{2} \)
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 \( 1 + (-3.78 - 3.78i)T + 53iT^{2} \)
59 \( 1 + (-4.48 - 7.77i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.275 + 0.476i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.71 + 6.38i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 6.29iT - 71T^{2} \)
73 \( 1 + (-6.89 + 6.89i)T - 73iT^{2} \)
79 \( 1 + (2.12 + 1.22i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.26 + 1.41i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 - 8.02T + 89T^{2} \)
97 \( 1 + (-2.59 + 0.695i)T + (84.0 - 48.5i)T^{2} \)
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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

−9.665315893202908772577213635654, −8.560511834825701023080311289000, −8.152641014810687459048067800010, −7.35252119413617948003871521908, −6.13182275510992802124702513199, −5.11466337564594820621284264635, −4.27366373680873791555490042977, −3.48033519344591326985306308707, −1.94240046236341097221552895254, −1.22273109299606084850941897173, 1.06456492687064453536324824026, 2.33463146568497828977979226002, 3.86026281970551408216106520860, 4.87192747832698746061636213582, 5.44038913996380249623938829809, 6.44015692678856335146611217441, 7.32712977752560170077680390718, 8.212748393311665950303310573230, 8.632374053809766515028592354145, 9.395426759745457689604679697409

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