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Properties

Label 2-1350-15.8-c1-0-20
Degree $2$
Conductor $1350$
Sign $0.899 + 0.437i$
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 + (3.34 − 3.34i)7-s + (−0.707 + 0.707i)8-s − 3i·11-s + (2.12 + 2.12i)13-s + 4.73·14-s − 1.00·16-s + (−5.79 − 5.79i)17-s − 4.19i·19-s + (2.12 − 2.12i)22-s + (2.12 − 2.12i)23-s + 3i·26-s + (3.34 + 3.34i)28-s − 2.19·29-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + 0.500i·4-s + (1.26 − 1.26i)7-s + (−0.250 + 0.250i)8-s − 0.904i·11-s + (0.588 + 0.588i)13-s + 1.26·14-s − 0.250·16-s + (−1.40 − 1.40i)17-s − 0.962i·19-s + (0.452 − 0.452i)22-s + (0.442 − 0.442i)23-s + 0.588i·26-s + (0.632 + 0.632i)28-s − 0.407·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 + 0.437i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.899 + 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $0.899 + 0.437i$
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: \((2,\ 1350,\ (\ :1/2),\ 0.899 + 0.437i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.337674364\)
\(L(\frac12)\) \(\approx\) \(2.337674364\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.34 + 3.34i)T - 7iT^{2} \)
11 \( 1 + 3iT - 11T^{2} \)
13 \( 1 + (-2.12 - 2.12i)T + 13iT^{2} \)
17 \( 1 + (5.79 + 5.79i)T + 17iT^{2} \)
19 \( 1 + 4.19iT - 19T^{2} \)
23 \( 1 + (-2.12 + 2.12i)T - 23iT^{2} \)
29 \( 1 + 2.19T + 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + (2.77 - 2.77i)T - 37iT^{2} \)
41 \( 1 + 6iT - 41T^{2} \)
43 \( 1 + (-5.13 - 5.13i)T + 43iT^{2} \)
47 \( 1 + (-6.36 - 6.36i)T + 47iT^{2} \)
53 \( 1 + (5.79 - 5.79i)T - 53iT^{2} \)
59 \( 1 - 7.39T + 59T^{2} \)
61 \( 1 - 1.19T + 61T^{2} \)
67 \( 1 + (-5.79 + 5.79i)T - 67iT^{2} \)
71 \( 1 + 11.1iT - 71T^{2} \)
73 \( 1 + (-10.9 - 10.9i)T + 73iT^{2} \)
79 \( 1 + 0.196iT - 79T^{2} \)
83 \( 1 + (11.5 - 11.5i)T - 83iT^{2} \)
89 \( 1 - 2.19T + 89T^{2} \)
97 \( 1 + (-10.3 + 10.3i)T - 97iT^{2} \)
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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

−9.225644216983346960915152086744, −8.705792134645983463927621830674, −7.73841341886724631311204838744, −7.07913957270081098152483400355, −6.39565648108158282813031539504, −5.12891068126206736691414974199, −4.55858392790488649627432629592, −3.74299674049081273310336159002, −2.41718873513721421987817455832, −0.846012620433440372408705899671, 1.68997328924518543236132507240, 2.22366744686821132209482183953, 3.66552266549810793932031855570, 4.54919492481527153378331911350, 5.45397800829502246105603803491, 6.01750981524960361758460853302, 7.21190434110785994851905755179, 8.327914849246099823685802366163, 8.718396400376564702335948586308, 9.745185529392979868508374643855

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