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Properties

Label 2-1350-9.4-c1-0-7
Degree $2$
Conductor $1350$
Sign $0.569 + 0.821i$
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 + (−2.22 − 3.85i)7-s + 0.999·8-s + (2.44 + 4.24i)11-s + (−2.22 + 3.85i)13-s + (−2.22 + 3.85i)14-s + (−0.5 − 0.866i)16-s + 4.89·17-s + 2.55·19-s + (2.44 − 4.24i)22-s + (1.22 − 2.12i)23-s + 4.44·26-s + 4.44·28-s + (1.22 + 2.12i)29-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.840 − 1.45i)7-s + 0.353·8-s + (0.738 + 1.27i)11-s + (−0.617 + 1.06i)13-s + (−0.594 + 1.02i)14-s + (−0.125 − 0.216i)16-s + 1.18·17-s + 0.585·19-s + (0.522 − 0.904i)22-s + (0.255 − 0.442i)23-s + 0.872·26-s + 0.840·28-s + (0.227 + 0.393i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.569 + 0.821i$
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: \((2,\ 1350,\ (\ :1/2),\ 0.569 + 0.821i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.240738212\)
\(L(\frac12)\) \(\approx\) \(1.240738212\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.22 + 3.85i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.44 - 4.24i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.22 - 3.85i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.89T + 17T^{2} \)
19 \( 1 - 2.55T + 19T^{2} \)
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 \( 1 + (-0.224 + 0.389i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.34T + 37T^{2} \)
41 \( 1 + (-4.5 + 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.72 + 6.45i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.550 + 0.953i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 8.44T + 53T^{2} \)
59 \( 1 + (0.275 - 0.476i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.17 + 12.4i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 1.34T + 71T^{2} \)
73 \( 1 + T + 73T^{2} \)
79 \( 1 + (-6.34 - 10.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.275 + 0.476i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 9T + 89T^{2} \)
97 \( 1 + (-5.39 - 9.35i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.674778385213991445660263291590, −9.029008683212354907317625142440, −7.67302773869769452407865516213, −7.15660355491176175714510682865, −6.52854642343283407729868194926, −4.98750045357892932365784600579, −4.10127385696797997119672722961, −3.44911574343316387898521412932, −2.06367613286116996381539154561, −0.828164511139725807232619152347, 0.921685271810782544767592998191, 2.75242108249649347458942398009, 3.43252292856135422807364686328, 5.03821614158340195656418184784, 5.88620968383546369561951242023, 6.13716080164507040364570553740, 7.39458497723224318478827294414, 8.183140778060607721203321770681, 8.887223907848463110384855290160, 9.617019009235990236945520510704

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