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Properties

Label 2-1350-15.8-c1-0-17
Degree $2$
Conductor $1350$
Sign $0.850 - 0.525i$
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 + (3.44 − 3.44i)7-s + (−0.707 + 0.707i)8-s + 4.56i·11-s + (−1.77 − 1.77i)13-s + 4.87·14-s − 1.00·16-s + (2.75 + 2.75i)17-s − 0.449i·19-s + (−3.22 + 3.22i)22-s + (5.90 − 5.90i)23-s − 2.51i·26-s + (3.44 + 3.44i)28-s + 0.317·29-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + 0.500i·4-s + (1.30 − 1.30i)7-s + (−0.250 + 0.250i)8-s + 1.37i·11-s + (−0.492 − 0.492i)13-s + 1.30·14-s − 0.250·16-s + (0.668 + 0.668i)17-s − 0.103i·19-s + (−0.687 + 0.687i)22-s + (1.23 − 1.23i)23-s − 0.492i·26-s + (0.651 + 0.651i)28-s + 0.0590·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.850 - 0.525i$
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: \((2,\ 1350,\ (\ :1/2),\ 0.850 - 0.525i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.563263856\)
\(L(\frac12)\) \(\approx\) \(2.563263856\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.44 + 3.44i)T - 7iT^{2} \)
11 \( 1 - 4.56iT - 11T^{2} \)
13 \( 1 + (1.77 + 1.77i)T + 13iT^{2} \)
17 \( 1 + (-2.75 - 2.75i)T + 17iT^{2} \)
19 \( 1 + 0.449iT - 19T^{2} \)
23 \( 1 + (-5.90 + 5.90i)T - 23iT^{2} \)
29 \( 1 - 0.317T + 29T^{2} \)
31 \( 1 - 3.44T + 31T^{2} \)
37 \( 1 - 37iT^{2} \)
41 \( 1 - 2.04iT - 41T^{2} \)
43 \( 1 + (-4.77 - 4.77i)T + 43iT^{2} \)
47 \( 1 + (-3.07 - 3.07i)T + 47iT^{2} \)
53 \( 1 + (-5.65 + 5.65i)T - 53iT^{2} \)
59 \( 1 - 2.82T + 59T^{2} \)
61 \( 1 + 3.55T + 61T^{2} \)
67 \( 1 + (2 - 2i)T - 67iT^{2} \)
71 \( 1 + 0.142iT - 71T^{2} \)
73 \( 1 + (0.449 + 0.449i)T + 73iT^{2} \)
79 \( 1 - 0.550iT - 79T^{2} \)
83 \( 1 + (6.75 - 6.75i)T - 83iT^{2} \)
89 \( 1 - 3.32T + 89T^{2} \)
97 \( 1 + (3.55 - 3.55i)T - 97iT^{2} \)
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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

−9.817380054161341261548280131336, −8.581161061900191782618436993280, −7.80184006927886075032675846617, −7.31408936667947928859990906447, −6.55106310921138249594103806337, −5.23052692502060860533007387812, −4.63944496760290371449274677923, −3.99052369707366229210106658570, −2.57055663309787019168099716981, −1.19145313703807799120895758381, 1.20530233060013913397589470786, 2.40696238748273358014855405772, 3.24549287318481807186564866046, 4.51434097545237328290402930504, 5.44898288695258589719027553178, 5.71853043297408259894186377855, 7.09061231568537524743821515194, 8.045559913273224081007414901230, 8.878423003125143197599836337853, 9.377639636856610530816641806819

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