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Properties

Label 2-1350-45.23-c1-0-14
Degree $2$
Conductor $1350$
Sign $0.537 + 0.843i$
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 + (0.707 − 0.707i)8-s + (−3 + 1.73i)11-s + (3.34 − 0.896i)13-s + (0.500 + 0.866i)16-s + (−4.24 − 4.24i)17-s − 5i·19-s + (−0.896 − 3.34i)22-s + (−1.55 − 5.79i)23-s + 3.46i·26-s + (3.46 + 6i)29-s + (−2 + 3.46i)31-s + (−0.965 + 0.258i)32-s + (5.19 − 3i)34-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (−0.433 − 0.249i)4-s + (0.249 − 0.249i)8-s + (−0.904 + 0.522i)11-s + (0.928 − 0.248i)13-s + (0.125 + 0.216i)16-s + (−1.02 − 1.02i)17-s − 1.14i·19-s + (−0.191 − 0.713i)22-s + (−0.323 − 1.20i)23-s + 0.679i·26-s + (0.643 + 1.11i)29-s + (−0.359 + 0.622i)31-s + (−0.170 + 0.0457i)32-s + (0.891 − 0.514i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.537 + 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.537 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.537 + 0.843i$
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: \((2,\ 1350,\ (\ :1/2),\ 0.537 + 0.843i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8788164490\)
\(L(\frac12)\) \(\approx\) \(0.8788164490\)
\(L(\frac{3}{2})\) 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 \( 1 + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (3 - 1.73i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.34 + 0.896i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (4.24 + 4.24i)T + 17iT^{2} \)
19 \( 1 + 5iT - 19T^{2} \)
23 \( 1 + (1.55 + 5.79i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-3.46 - 6i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.89 - 4.89i)T - 37iT^{2} \)
41 \( 1 + (-1.5 - 0.866i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.13 + 11.7i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-1.55 + 5.79i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-4.24 + 4.24i)T - 53iT^{2} \)
59 \( 1 + (0.866 - 1.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.24 + 8.36i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 6.92iT - 71T^{2} \)
73 \( 1 + (8.57 + 8.57i)T + 73iT^{2} \)
79 \( 1 + (-12.1 + 7i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.69 - 2.32i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 - 12.1T + 89T^{2} \)
97 \( 1 + (-5.01 - 1.34i)T + (84.0 + 48.5i)T^{2} \)
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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

−9.111131546146960896808532892390, −8.771274967317130688248608368855, −7.83011921011230002570394578401, −6.92517134787697015368755049070, −6.44538714063074558911738872155, −5.11198249127094848023591878290, −4.76896603818387863779437275632, −3.37102128508124179314405231337, −2.17184774204511474164545603884, −0.39549994904944847830084005680, 1.37841394837615382744794933492, 2.49816346797713282965529281664, 3.66075578043579037988752687398, 4.33801283632997244411210065913, 5.69227125504992882350803230249, 6.21019722484517981741393524887, 7.63556085194914717132262358059, 8.189068438101343712570800140840, 8.975240209486102795401749395844, 9.818351686902663873461742586219

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