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Properties

Label 2-1350-45.2-c1-0-5
Degree $2$
Conductor $1350$
Sign $0.860 - 0.509i$
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 + (−2.32 + 0.622i)7-s + (0.707 + 0.707i)8-s + (−0.991 − 0.572i)11-s + (−2.38 − 0.640i)13-s + (1.20 + 2.08i)14-s + (0.500 − 0.866i)16-s + (4.99 − 4.99i)17-s + 2.78i·19-s + (−0.296 + 1.10i)22-s + (−1.59 + 5.95i)23-s + 2.47i·26-s + (1.70 − 1.70i)28-s + (0.672 − 1.16i)29-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (−0.433 + 0.249i)4-s + (−0.877 + 0.235i)7-s + (0.249 + 0.249i)8-s + (−0.299 − 0.172i)11-s + (−0.662 − 0.177i)13-s + (0.321 + 0.556i)14-s + (0.125 − 0.216i)16-s + (1.21 − 1.21i)17-s + 0.638i·19-s + (−0.0631 + 0.235i)22-s + (−0.332 + 1.24i)23-s + 0.485i·26-s + (0.321 − 0.321i)28-s + (0.124 − 0.216i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 - 0.509i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.860 - 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.860 - 0.509i$
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: \((2,\ 1350,\ (\ :1/2),\ 0.860 - 0.509i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9459674438\)
\(L(\frac12)\) \(\approx\) \(0.9459674438\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.32 - 0.622i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (0.991 + 0.572i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.38 + 0.640i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (-4.99 + 4.99i)T - 17iT^{2} \)
19 \( 1 - 2.78iT - 19T^{2} \)
23 \( 1 + (1.59 - 5.95i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-0.672 + 1.16i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.25 - 2.16i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-8.16 - 8.16i)T + 37iT^{2} \)
41 \( 1 + (-1.70 + 0.986i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.32 - 8.68i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-3.19 - 11.9i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.84 + 1.84i)T + 53iT^{2} \)
59 \( 1 + (-1.31 - 2.27i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.54 - 6.13i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.0146 + 0.0545i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 9.10iT - 71T^{2} \)
73 \( 1 + (7.82 - 7.82i)T - 73iT^{2} \)
79 \( 1 + (-8.46 - 4.88i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.70 + 0.724i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + 4.87T + 89T^{2} \)
97 \( 1 + (-7.79 + 2.08i)T + (84.0 - 48.5i)T^{2} \)
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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

−9.688251534299028017907538190193, −9.267107839731366454252046040096, −7.950276086072170482530541634601, −7.54851864047555482366837985062, −6.27710220793516748500553527978, −5.46593999309944969172829061271, −4.47776180380717638367472034957, −3.22004124741842443206339374284, −2.73576362122136440678106740172, −1.13212289046043470549965188524, 0.47822492912447458098952994336, 2.28797260901112600099034218567, 3.56708114183726810105204356646, 4.48965580996614161471846367626, 5.56166611730344456126331541076, 6.29435128650414812665841789704, 7.10813505235774209302195662230, 7.82519513500026981190937770052, 8.672442945699716128028137844949, 9.533468267629635103696634587636

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