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Properties

Label 2-1350-15.2-c1-0-11
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: $\chi_{1350} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1350,\ (\ :1/2),\ 0.899 - 0.437i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.584522240\)
\(L(\frac12)\) \(\approx\) \(1.584522240\)
\(L(\frac{3}{2})\) 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

−9.391993222377961395471073402412, −8.760289304007869921122670326295, −8.022548005743664972111090492990, −7.60174745324129962607101796305, −6.18240839443341804819261707989, −5.56595803043379177316722720839, −5.00521407404385390054255965253, −3.49810572796820094086133146295, −2.30041725040024388137683337834, −1.00604760263795794284100183661, 1.16368561808638211248909252700, 1.90163123909653851049432124986, 3.52972974464268876092933255015, 4.27101303787352157446590390760, 5.10888064010601982048987353123, 6.47988517698564601029681080456, 7.37084850158716834388487827373, 7.981906578650254677968820028255, 8.641359026437862300530602777314, 9.824907303246309230692394832753

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