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Properties

Label 2-1350-15.8-c1-0-13
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: $\chi_{1350} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1350,\ (\ :1/2),\ 0.608 - 0.793i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.362428523\)
\(L(\frac12)\) \(\approx\) \(2.362428523\)
\(L(\frac{3}{2})\) 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

−9.604884774976804766809899471822, −8.707125219708802364284066366101, −8.033989100128959887500083463250, −7.31551710150515253376098004627, −6.13556485441212332073409089680, −5.85218314354954895972419322083, −4.53383758794422942311715556390, −3.87864838548261594036174125617, −2.80367136786194763605564137039, −1.23168300996773421820606583965, 1.03215239411056794736302919822, 2.38587209817188255723455286720, 3.22356086965437554225874724068, 4.50144347718130977111480214734, 5.05272393987701445436225506863, 6.04857218802525074015440744416, 6.97104653949195721481437466291, 7.84842246611426849832338226729, 8.859057139677870993067102707336, 9.532546550954550140215461451099

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