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Properties

Label 2-1350-45.32-c1-0-14
Degree $2$
Conductor $1350$
Sign $0.913 + 0.407i$
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 + (1.5 − 0.866i)11-s + (−1.79 − 6.69i)13-s + (0.500 + 0.866i)16-s + (2.12 − 2.12i)17-s − 4i·19-s + (1.67 − 0.448i)22-s + (5.79 − 1.55i)23-s − 6.92i·26-s + (1.73 + 3i)29-s + (−2 + 3.46i)31-s + (0.258 + 0.965i)32-s + (2.59 − 1.5i)34-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (0.433 + 0.249i)4-s + (0.249 + 0.249i)8-s + (0.452 − 0.261i)11-s + (−0.497 − 1.85i)13-s + (0.125 + 0.216i)16-s + (0.514 − 0.514i)17-s − 0.917i·19-s + (0.356 − 0.0955i)22-s + (1.20 − 0.323i)23-s − 1.35i·26-s + (0.321 + 0.557i)29-s + (−0.359 + 0.622i)31-s + (0.0457 + 0.170i)32-s + (0.445 − 0.257i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 + 0.407i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.913 + 0.407i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.913 + 0.407i$
Analytic conductor: \(10.7798\)
Root analytic conductor: \(3.28326\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1350} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1350,\ (\ :1/2),\ 0.913 + 0.407i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.581365768\)
\(L(\frac12)\) \(\approx\) \(2.581365768\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 + 0.866i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.79 + 6.69i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (-2.12 + 2.12i)T - 17iT^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + (-5.79 + 1.55i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-1.73 - 3i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.89 - 4.89i)T + 37iT^{2} \)
41 \( 1 + (-6 - 3.46i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (8.36 + 2.24i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-11.5 - 3.10i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (8.48 + 8.48i)T + 53iT^{2} \)
59 \( 1 + (4.33 - 7.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.34 - 0.896i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 3.46iT - 71T^{2} \)
73 \( 1 + (-4.89 + 4.89i)T - 73iT^{2} \)
79 \( 1 + (-3.46 + 2i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.32 + 8.69i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + 1.73T + 89T^{2} \)
97 \( 1 + (-4.03 + 15.0i)T + (-84.0 - 48.5i)T^{2} \)
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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

−9.541355737177464523077917398618, −8.672082223009893816065037526869, −7.75157445352009272002768304013, −7.09232916182077252045462630145, −6.14928024727211321810981990423, −5.25149498324785092658375942341, −4.66206539009355371446905469383, −3.26505955360919054124322772418, −2.76815434471160039885319125454, −0.937856155063854380064636364261, 1.46199607149745418142887689695, 2.47951830024708823576546083451, 3.83283042588742194556316194249, 4.35734087313443383930056255506, 5.44796441835116009432376673666, 6.31201267855521923735988308256, 7.07768202545864391097347777478, 7.86425901056988832122361274492, 9.133766080739873216869112470816, 9.547635406599191271947038492050

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