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Properties

Label 2-1350-9.4-c1-0-14
Degree $2$
Conductor $1350$
Sign $-0.964 + 0.263i$
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  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.224 + 0.389i)7-s + 0.999·8-s + (−2.44 − 4.24i)11-s + (0.224 − 0.389i)13-s + (0.224 − 0.389i)14-s + (−0.5 − 0.866i)16-s − 4.89·17-s + 7.44·19-s + (−2.44 + 4.24i)22-s + (−1.22 + 2.12i)23-s − 0.449·26-s − 0.449·28-s + (−1.22 − 2.12i)29-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.0849 + 0.147i)7-s + 0.353·8-s + (−0.738 − 1.27i)11-s + (0.0623 − 0.107i)13-s + (0.0600 − 0.104i)14-s + (−0.125 − 0.216i)16-s − 1.18·17-s + 1.70·19-s + (−0.522 + 0.904i)22-s + (−0.255 + 0.442i)23-s − 0.0881·26-s − 0.0849·28-s + (−0.227 − 0.393i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.263i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 + 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $-0.964 + 0.263i$
Analytic conductor: \(10.7798\)
Root analytic conductor: \(3.28326\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1350} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1350,\ (\ :1/2),\ -0.964 + 0.263i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6431069890\)
\(L(\frac12)\) \(\approx\) \(0.6431069890\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-0.224 - 0.389i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.44 + 4.24i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.224 + 0.389i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.89T + 17T^{2} \)
19 \( 1 - 7.44T + 19T^{2} \)
23 \( 1 + (1.22 - 2.12i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.22 + 2.12i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.22 - 3.85i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 11.3T + 37T^{2} \)
41 \( 1 + (-4.5 + 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.27 + 2.20i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.44 + 9.43i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 3.55T + 53T^{2} \)
59 \( 1 + (2.72 - 4.71i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.174 - 0.301i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 13.3T + 71T^{2} \)
73 \( 1 + T + 73T^{2} \)
79 \( 1 + (8.34 + 14.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.72 + 4.71i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 9T + 89T^{2} \)
97 \( 1 + (4.39 + 7.61i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.064566985844949311956442668575, −8.707261015608348437970277270385, −7.74257239822342672934924955059, −6.97554314916157853621775659179, −5.71106638383866389712650997756, −5.09101543743256552577129754126, −3.72797950645469542711784595591, −3.01050848999594023747254095874, −1.81132451814986329890437104502, −0.29394151302254968708891362939, 1.54982909476253425524237356449, 2.81172627278520045367829111302, 4.27833021072743980906953713822, 4.97328900752775700471925369182, 5.89015824158221458873918492438, 6.96469164746281007225269853615, 7.43988876861362734276738943038, 8.247927800237903223031388501151, 9.244511682340036016994301044349, 9.763469355285350961787473480090

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