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Properties

Label 2-1350-45.4-c1-0-3
Degree $2$
Conductor $1350$
Sign $-0.232 - 0.972i$
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  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.866 + 0.5i)7-s + 0.999i·8-s + (3 + 5.19i)11-s + (1.73 + i)13-s + (0.499 − 0.866i)14-s + (−0.5 − 0.866i)16-s + 4·19-s + (−5.19 − 3i)22-s + (−7.79 − 4.5i)23-s − 1.99·26-s + 0.999i·28-s + (−1.5 − 2.59i)29-s + (2 − 3.46i)31-s + (0.866 + 0.499i)32-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.327 + 0.188i)7-s + 0.353i·8-s + (0.904 + 1.56i)11-s + (0.480 + 0.277i)13-s + (0.133 − 0.231i)14-s + (−0.125 − 0.216i)16-s + 0.917·19-s + (−1.10 − 0.639i)22-s + (−1.62 − 0.938i)23-s − 0.392·26-s + 0.188i·28-s + (−0.278 − 0.482i)29-s + (0.359 − 0.622i)31-s + (0.153 + 0.0883i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.232 - 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.232 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $-0.232 - 0.972i$
Analytic conductor: \(10.7798\)
Root analytic conductor: \(3.28326\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1350} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1350,\ (\ :1/2),\ -0.232 - 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.051445710\)
\(L(\frac12)\) \(\approx\) \(1.051445710\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (0.866 - 0.5i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3 - 5.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.73 - i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + (7.79 + 4.5i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 8iT - 37T^{2} \)
41 \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.92 - 4i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.59 + 1.5i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + (3 - 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.5 - 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-11.2 - 6.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 - 4iT - 73T^{2} \)
79 \( 1 + (5 + 8.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.79 - 4.5i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 9T + 89T^{2} \)
97 \( 1 + (-1.73 + i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.910400464598619480530995882419, −9.090160258331437215715667023235, −8.226737196874860338104981778063, −7.42825534093253100965342392401, −6.58721246122946685593871398548, −6.01834705771960148191738975580, −4.75886360451060189161550512102, −3.93076586080313024974586665576, −2.46767941547755616025477578686, −1.35725628447687316168321350547, 0.57219697366140985311191121789, 1.82085486482291096182027190950, 3.48904186335533102719937067109, 3.60285602823852269595081708751, 5.33468050562385146857043572243, 6.15685404959915365928806921668, 6.97193588989761362503316452261, 8.009759664972764792256778531339, 8.588044629411438715620214104157, 9.422761651759154471040254413183

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