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Properties

Label 2-1350-15.8-c1-0-0
Degree $2$
Conductor $1350$
Sign $-0.608 + 0.793i$
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  + (0.707 + 0.707i)2-s + 1.00i·4-s + (−3.34 + 3.34i)7-s + (−0.707 + 0.707i)8-s + 3i·11-s + (−2.12 − 2.12i)13-s − 4.73·14-s − 1.00·16-s + (−5.79 − 5.79i)17-s − 4.19i·19-s + (−2.12 + 2.12i)22-s + (2.12 − 2.12i)23-s − 3i·26-s + (−3.34 − 3.34i)28-s + 2.19·29-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + 0.500i·4-s + (−1.26 + 1.26i)7-s + (−0.250 + 0.250i)8-s + 0.904i·11-s + (−0.588 − 0.588i)13-s − 1.26·14-s − 0.250·16-s + (−1.40 − 1.40i)17-s − 0.962i·19-s + (−0.452 + 0.452i)22-s + (0.442 − 0.442i)23-s − 0.588i·26-s + (−0.632 − 0.632i)28-s + 0.407·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.608 + 0.793i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $-0.608 + 0.793i$
Analytic conductor: \(10.7798\)
Root analytic conductor: \(3.28326\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1350} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1350,\ (\ :1/2),\ -0.608 + 0.793i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2086063103\)
\(L(\frac12)\) \(\approx\) \(0.2086063103\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (3.34 - 3.34i)T - 7iT^{2} \)
11 \( 1 - 3iT - 11T^{2} \)
13 \( 1 + (2.12 + 2.12i)T + 13iT^{2} \)
17 \( 1 + (5.79 + 5.79i)T + 17iT^{2} \)
19 \( 1 + 4.19iT - 19T^{2} \)
23 \( 1 + (-2.12 + 2.12i)T - 23iT^{2} \)
29 \( 1 - 2.19T + 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + (-2.77 + 2.77i)T - 37iT^{2} \)
41 \( 1 - 6iT - 41T^{2} \)
43 \( 1 + (5.13 + 5.13i)T + 43iT^{2} \)
47 \( 1 + (-6.36 - 6.36i)T + 47iT^{2} \)
53 \( 1 + (5.79 - 5.79i)T - 53iT^{2} \)
59 \( 1 + 7.39T + 59T^{2} \)
61 \( 1 - 1.19T + 61T^{2} \)
67 \( 1 + (5.79 - 5.79i)T - 67iT^{2} \)
71 \( 1 - 11.1iT - 71T^{2} \)
73 \( 1 + (10.9 + 10.9i)T + 73iT^{2} \)
79 \( 1 + 0.196iT - 79T^{2} \)
83 \( 1 + (11.5 - 11.5i)T - 83iT^{2} \)
89 \( 1 + 2.19T + 89T^{2} \)
97 \( 1 + (10.3 - 10.3i)T - 97iT^{2} \)
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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

−9.823742791395423589290214488624, −9.245582689754114913274600463772, −8.671992537928036619312989419987, −7.36482950530375529920759759494, −6.84371761192166389113219388107, −6.04213272577171185452708647704, −5.09787688405241823142768079267, −4.43059803439000731204936558269, −2.85513042069954133475002569549, −2.55823350723252009233420124904, 0.06732483289705444293188508886, 1.63183200362910665536870393624, 3.05959634284118305548232894144, 3.82321686734096356563585738946, 4.49914414098809104712841364701, 5.87097847313627005533592728872, 6.49916107574069671156184622182, 7.22492827293225604543926562087, 8.382578577377982195809361127830, 9.282126722712161541435376868071

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