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Properties

Label 2-1350-45.2-c1-0-14
Degree $2$
Conductor $1350$
Sign $0.912 + 0.408i$
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 + (2.56 − 0.686i)7-s + (−0.707 − 0.707i)8-s + (−4.15 − 2.39i)11-s + (−0.581 − 0.155i)13-s + (1.32 + 2.29i)14-s + (0.500 − 0.866i)16-s + (4.40 − 4.40i)17-s − 5.19i·19-s + (1.24 − 4.63i)22-s + (−0.681 + 2.54i)23-s − 0.602i·26-s + (−1.87 + 1.87i)28-s + (0.920 − 1.59i)29-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (−0.433 + 0.249i)4-s + (0.968 − 0.259i)7-s + (−0.249 − 0.249i)8-s + (−1.25 − 0.723i)11-s + (−0.161 − 0.0432i)13-s + (0.354 + 0.613i)14-s + (0.125 − 0.216i)16-s + (1.06 − 1.06i)17-s − 1.19i·19-s + (0.264 − 0.988i)22-s + (−0.142 + 0.530i)23-s − 0.118i·26-s + (−0.354 + 0.354i)28-s + (0.170 − 0.295i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 + 0.408i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.912 + 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.912 + 0.408i$
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: \((2,\ 1350,\ (\ :1/2),\ 0.912 + 0.408i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.588928357\)
\(L(\frac12)\) \(\approx\) \(1.588928357\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.56 + 0.686i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (4.15 + 2.39i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.581 + 0.155i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (-4.40 + 4.40i)T - 17iT^{2} \)
19 \( 1 + 5.19iT - 19T^{2} \)
23 \( 1 + (0.681 - 2.54i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-0.920 + 1.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.03 + 3.53i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.632 + 0.632i)T + 37iT^{2} \)
41 \( 1 + (-5.58 + 3.22i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.644 - 2.40i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (1.02 + 3.82i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.31 + 1.31i)T + 53iT^{2} \)
59 \( 1 + (-0.0645 - 0.111i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.27 + 10.8i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.85 - 10.6i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 10.4iT - 71T^{2} \)
73 \( 1 + (3.30 - 3.30i)T - 73iT^{2} \)
79 \( 1 + (-3.62 - 2.09i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-11.1 + 2.97i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + 2.04T + 89T^{2} \)
97 \( 1 + (-16.7 + 4.47i)T + (84.0 - 48.5i)T^{2} \)
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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

−9.424322606050521673505616302084, −8.557803226699099416002855098041, −7.64071343309949461471557376848, −7.47301794032934310026410762594, −6.15799287783629876723948134854, −5.22157158693469436807389283231, −4.83876321512718182788871887719, −3.52266071089163215614019142738, −2.44950147761049528235079171728, −0.64526190204718728026625388036, 1.44459638839284162398055664314, 2.35237950066211493283990519143, 3.51951165551942973944421818791, 4.58598311396594065205350266600, 5.29458474224201496614990408458, 6.09415415012742429755945154522, 7.54467879047008363267475902117, 8.036940974154052949688386343178, 8.835305791256833840322447538006, 10.03825543595384141927614862358

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