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Properties

Label 2-1350-9.4-c1-0-15
Degree $2$
Conductor $1350$
Sign $-0.939 - 0.342i$
Analytic cond. $10.7798$
Root an. cond. $3.28326$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (1 + 1.73i)7-s + 0.999·8-s + (−2 + 3.46i)13-s + (0.999 − 1.73i)14-s + (−0.5 − 0.866i)16-s − 6·17-s − 7·19-s + 3.99·26-s − 1.99·28-s + (−3 − 5.19i)29-s + (5 − 8.66i)31-s + (−0.499 + 0.866i)32-s + (3 + 5.19i)34-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.377 + 0.654i)7-s + 0.353·8-s + (−0.554 + 0.960i)13-s + (0.267 − 0.462i)14-s + (−0.125 − 0.216i)16-s − 1.45·17-s − 1.60·19-s + 0.784·26-s − 0.377·28-s + (−0.557 − 0.964i)29-s + (0.898 − 1.55i)31-s + (−0.0883 + 0.153i)32-s + (0.514 + 0.891i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $-0.939 - 0.342i$
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: \(1\)
Selberg data: \((2,\ 1350,\ (\ :1/2),\ -0.939 - 0.342i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - 1.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2 - 3.46i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 + 7T + 19T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5 + 8.66i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + (-4.5 + 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 12T + 53T^{2} \)
59 \( 1 + (4.5 - 7.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2 - 3.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.5 - 11.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 - T + 73T^{2} \)
79 \( 1 + (1 + 1.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.5 + 7.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 15T + 89T^{2} \)
97 \( 1 + (-8.5 - 14.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.037593995785764802196392734289, −8.671884453741680039544031929574, −7.70251703631663084625827794272, −6.72601532554536440988638821294, −5.88242330502159593046046638834, −4.56841762651614364130230098575, −4.10255096288302908897523636584, −2.44540202341530139031734496939, −2.00350616689028588710931413556, 0, 1.60962770375240817886116176686, 3.00594619940090116148398393944, 4.43422675380092256601168355479, 4.90189932926467590833400999713, 6.18403561968746032644569003753, 6.78808972436785101388940283971, 7.68442796592499615592652008523, 8.363251470296773428563682808748, 9.079968569024001693247739654259

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