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Properties

Label 2-1350-15.2-c1-0-17
Degree $2$
Conductor $1350$
Sign $-0.525 + 0.850i$
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 + (−1.36 − 1.36i)7-s + (−0.707 − 0.707i)8-s + 2.31i·11-s + (3.46 − 3.46i)13-s − 1.93·14-s − 1.00·16-s + (3.86 − 3.86i)17-s + 0.535i·19-s + (1.63 + 1.63i)22-s + (−1.41 − 1.41i)23-s − 4.89i·26-s + (−1.36 + 1.36i)28-s + 3.48·29-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s − 0.500i·4-s + (−0.516 − 0.516i)7-s + (−0.250 − 0.250i)8-s + 0.696i·11-s + (0.960 − 0.960i)13-s − 0.516·14-s − 0.250·16-s + (0.937 − 0.937i)17-s + 0.122i·19-s + (0.348 + 0.348i)22-s + (−0.294 − 0.294i)23-s − 0.960i·26-s + (−0.258 + 0.258i)28-s + 0.647·29-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(1.828593488\)
\(L(\frac12)\) \(\approx\) \(1.828593488\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.36 + 1.36i)T + 7iT^{2} \)
11 \( 1 - 2.31iT - 11T^{2} \)
13 \( 1 + (-3.46 + 3.46i)T - 13iT^{2} \)
17 \( 1 + (-3.86 + 3.86i)T - 17iT^{2} \)
19 \( 1 - 0.535iT - 19T^{2} \)
23 \( 1 + (1.41 + 1.41i)T + 23iT^{2} \)
29 \( 1 - 3.48T + 29T^{2} \)
31 \( 1 + 10.6T + 31T^{2} \)
37 \( 1 + (8.19 + 8.19i)T + 37iT^{2} \)
41 \( 1 + 9.52iT - 41T^{2} \)
43 \( 1 + (-4.73 + 4.73i)T - 43iT^{2} \)
47 \( 1 + (4.24 - 4.24i)T - 47iT^{2} \)
53 \( 1 + (2.26 + 2.26i)T + 53iT^{2} \)
59 \( 1 - 10.1T + 59T^{2} \)
61 \( 1 - 2.53T + 61T^{2} \)
67 \( 1 + (4.19 + 4.19i)T + 67iT^{2} \)
71 \( 1 - 4.14iT - 71T^{2} \)
73 \( 1 + (7.63 - 7.63i)T - 73iT^{2} \)
79 \( 1 + 3.46iT - 79T^{2} \)
83 \( 1 + (-11.2 - 11.2i)T + 83iT^{2} \)
89 \( 1 - 8.76T + 89T^{2} \)
97 \( 1 + (-1.43 - 1.43i)T + 97iT^{2} \)
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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

−9.508853197089872709022924995190, −8.673790625025038650612294590223, −7.52255824970918817274615554552, −6.92422273384941324767400061684, −5.75913977072125209121629328796, −5.18141895960205747401282158805, −3.88569984962702533978880007744, −3.36785063954533609870521789446, −2.08195279556612348396631170275, −0.64834530101153054332666186568, 1.61114214320322358257257355579, 3.15267955399192766231155273181, 3.76264399209538606951995578161, 4.92546477534959295896252998248, 5.98911886883933660248471170276, 6.28045746726682156444481914464, 7.34718560399943356946094057567, 8.335232496106725255393517809722, 8.857569276321512908298713604776, 9.753690386770611068876913466123

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