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Properties

Label 2-1350-45.38-c1-0-2
Degree $2$
Conductor $1350$
Sign $0.300 - 0.953i$
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  + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (1.22 + 4.57i)7-s + (0.707 − 0.707i)8-s + (−3 − 1.73i)11-s + (−1.22 + 4.57i)13-s + (2.36 + 4.09i)14-s + (0.500 − 0.866i)16-s + 3.19i·19-s + (−3.34 − 0.896i)22-s + (2.12 + 0.568i)23-s + 4.73i·26-s + (3.34 + 3.34i)28-s + (−5.36 + 9.29i)29-s + (−0.0980 − 0.169i)31-s + (0.258 − 0.965i)32-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (0.433 − 0.249i)4-s + (0.462 + 1.72i)7-s + (0.249 − 0.249i)8-s + (−0.904 − 0.522i)11-s + (−0.339 + 1.26i)13-s + (0.632 + 1.09i)14-s + (0.125 − 0.216i)16-s + 0.733i·19-s + (−0.713 − 0.191i)22-s + (0.442 + 0.118i)23-s + 0.928i·26-s + (0.632 + 0.632i)28-s + (−0.996 + 1.72i)29-s + (−0.0176 − 0.0305i)31-s + (0.0457 − 0.170i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.300 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.300 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.300 - 0.953i$
Analytic conductor: \(10.7798\)
Root analytic conductor: \(3.28326\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1350} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1350,\ (\ :1/2),\ 0.300 - 0.953i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.232703426\)
\(L(\frac12)\) \(\approx\) \(2.232703426\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 + 0.258i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-1.22 - 4.57i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (3 + 1.73i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.22 - 4.57i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + 17iT^{2} \)
19 \( 1 - 3.19iT - 19T^{2} \)
23 \( 1 + (-2.12 - 0.568i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (5.36 - 9.29i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.0980 + 0.169i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.79 + 5.79i)T - 37iT^{2} \)
41 \( 1 + (-1.5 + 0.866i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.448 + 0.120i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (5.79 - 1.55i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-5.79 + 5.79i)T - 53iT^{2} \)
59 \( 1 + (-2.76 - 4.79i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2 + 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.34 - 1.43i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 7.26iT - 71T^{2} \)
73 \( 1 + (3.67 + 3.67i)T + 73iT^{2} \)
79 \( 1 + (-8.66 - 5i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.45 - 16.6i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 - 8.66T + 89T^{2} \)
97 \( 1 + (-0.688 - 2.56i)T + (-84.0 + 48.5i)T^{2} \)
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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

−9.656530026789321998914327225092, −8.995071134733168498795773907816, −8.225520690127318166956466994401, −7.25589312725872908002029239695, −6.20953269566316089562298584324, −5.44131974598869267204987916232, −4.93158173510881551107388159908, −3.66083152309126161565373802502, −2.57671025888676560858792009065, −1.81785473716543243175123694840, 0.70882408511497433174964790736, 2.36156854164168235709159158059, 3.45192756336177368653909445784, 4.46918835980431177003424945307, 5.01200774998653271613085433364, 6.07072266807902821333136081972, 7.18613337799250549203607738534, 7.61097479204661778426718509322, 8.248011300593553046394136564036, 9.749766824807364522617896307859

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