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Properties

Label 2-1350-15.2-c1-0-15
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 + (0.896 + 0.896i)7-s + (0.707 + 0.707i)8-s − 3i·11-s + (2.12 − 2.12i)13-s − 1.26·14-s − 1.00·16-s + (−1.55 + 1.55i)17-s − 6.19i·19-s + (2.12 + 2.12i)22-s + (−2.12 − 2.12i)23-s + 3i·26-s + (0.896 − 0.896i)28-s − 8.19·29-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (0.338 + 0.338i)7-s + (0.250 + 0.250i)8-s − 0.904i·11-s + (0.588 − 0.588i)13-s − 0.338·14-s − 0.250·16-s + (−0.376 + 0.376i)17-s − 1.42i·19-s + (0.452 + 0.452i)22-s + (−0.442 − 0.442i)23-s + 0.588i·26-s + (0.169 − 0.169i)28-s − 1.52·29-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: $\chi_{1350} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1350,\ (\ :1/2),\ 0.608 + 0.793i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.063091114\)
\(L(\frac12)\) \(\approx\) \(1.063091114\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.896 - 0.896i)T + 7iT^{2} \)
11 \( 1 + 3iT - 11T^{2} \)
13 \( 1 + (-2.12 + 2.12i)T - 13iT^{2} \)
17 \( 1 + (1.55 - 1.55i)T - 17iT^{2} \)
19 \( 1 + 6.19iT - 19T^{2} \)
23 \( 1 + (2.12 + 2.12i)T + 23iT^{2} \)
29 \( 1 + 8.19T + 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + (-7.02 - 7.02i)T + 37iT^{2} \)
41 \( 1 + 6iT - 41T^{2} \)
43 \( 1 + (-7.58 + 7.58i)T - 43iT^{2} \)
47 \( 1 + (6.36 - 6.36i)T - 47iT^{2} \)
53 \( 1 + (1.55 + 1.55i)T + 53iT^{2} \)
59 \( 1 - 13.3T + 59T^{2} \)
61 \( 1 + 9.19T + 61T^{2} \)
67 \( 1 + (1.55 + 1.55i)T + 67iT^{2} \)
71 \( 1 + 0.803iT - 71T^{2} \)
73 \( 1 + (-6.03 + 6.03i)T - 73iT^{2} \)
79 \( 1 + 10.1iT - 79T^{2} \)
83 \( 1 + (3.10 + 3.10i)T + 83iT^{2} \)
89 \( 1 - 8.19T + 89T^{2} \)
97 \( 1 + (1.88 + 1.88i)T + 97iT^{2} \)
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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

−9.209840672968511034700763846601, −8.681977600822778502233185994896, −7.993476278119332052353142210911, −7.11640058943353434235265913545, −6.14339681641352935181771131464, −5.56106862724367152994134598283, −4.51833731289144488279841802425, −3.30882528195806494555472071147, −2.05410301838966525042489425367, −0.54242050434655473379279638548, 1.37629595288804470575697488086, 2.28088967688965539582230508217, 3.73642977488491913422420120324, 4.32616727665330000030499225190, 5.56327617868676416663734056207, 6.57387243147352117944930640091, 7.58969153828471736454559839989, 7.991754322337710405177645027959, 9.211167318467304992941910061115, 9.594088760668680366669842124382

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