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Properties

Label 2-1350-45.38-c1-0-13
Degree $2$
Conductor $1350$
Sign $-0.103 + 0.994i$
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 + (0.707 − 0.707i)8-s + (−3 − 1.73i)11-s + (0.896 − 3.34i)13-s + (0.500 − 0.866i)16-s + (−4.24 − 4.24i)17-s − 5i·19-s + (−3.34 − 0.896i)22-s + (5.79 + 1.55i)23-s − 3.46i·26-s + (−3.46 + 6i)29-s + (−2 − 3.46i)31-s + (0.258 − 0.965i)32-s + (−5.19 − 3i)34-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (0.433 − 0.249i)4-s + (0.249 − 0.249i)8-s + (−0.904 − 0.522i)11-s + (0.248 − 0.928i)13-s + (0.125 − 0.216i)16-s + (−1.02 − 1.02i)17-s − 1.14i·19-s + (−0.713 − 0.191i)22-s + (1.20 + 0.323i)23-s − 0.679i·26-s + (−0.643 + 1.11i)29-s + (−0.359 − 0.622i)31-s + (0.0457 − 0.170i)32-s + (−0.891 − 0.514i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.103 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.103 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $-0.103 + 0.994i$
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: \((2,\ 1350,\ (\ :1/2),\ -0.103 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.086467767\)
\(L(\frac12)\) \(\approx\) \(2.086467767\)
\(L(\frac{3}{2})\) 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 \( 1 + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (3 + 1.73i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.896 + 3.34i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (4.24 + 4.24i)T + 17iT^{2} \)
19 \( 1 + 5iT - 19T^{2} \)
23 \( 1 + (-5.79 - 1.55i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (3.46 - 6i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.89 + 4.89i)T - 37iT^{2} \)
41 \( 1 + (-1.5 + 0.866i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-11.7 + 3.13i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (5.79 - 1.55i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-4.24 + 4.24i)T - 53iT^{2} \)
59 \( 1 + (-0.866 - 1.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.36 + 2.24i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 6.92iT - 71T^{2} \)
73 \( 1 + (-8.57 - 8.57i)T + 73iT^{2} \)
79 \( 1 + (12.1 + 7i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.32 + 8.69i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + 12.1T + 89T^{2} \)
97 \( 1 + (-1.34 - 5.01i)T + (-84.0 + 48.5i)T^{2} \)
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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

−9.309466652253693522646859971675, −8.723484337797344360272433527670, −7.49119567021641667867284354294, −7.03260433197740611331393795975, −5.76596765206947335638328827885, −5.24907430295319563579935051710, −4.30357818853498163938347574844, −3.09186167636952181843603397234, −2.45823413536895443069158487152, −0.66273122598204034820750098419, 1.73160998500716616965290274855, 2.75766177741297710221695946329, 4.03315511475874063734508270302, 4.60489379175595065054744773586, 5.72141520730176275927768274256, 6.40536915811127936855140549149, 7.30632722900081802940413780758, 8.072549802813462084055224840364, 8.944264351967571788290896014445, 9.864447663576327764299569864878

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