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Properties

Label 2-1350-15.2-c1-0-2
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 + (−0.707 − 0.707i)8-s + 5.19i·11-s + (−3.67 + 3.67i)13-s − 1.00·16-s + (−2.12 + 2.12i)17-s + 2i·19-s + (3.67 + 3.67i)22-s + (2.12 + 2.12i)23-s + 5.19i·26-s + 5.19·29-s − 5·31-s + (−0.707 + 0.707i)32-s + 3i·34-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s − 0.500i·4-s + (−0.250 − 0.250i)8-s + 1.56i·11-s + (−1.01 + 1.01i)13-s − 0.250·16-s + (−0.514 + 0.514i)17-s + 0.458i·19-s + (0.783 + 0.783i)22-s + (0.442 + 0.442i)23-s + 1.01i·26-s + 0.964·29-s − 0.898·31-s + (−0.125 + 0.125i)32-s + 0.514i·34-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

\(L(1)\) \(\approx\) \(1.539615499\)
\(L(\frac12)\) \(\approx\) \(1.539615499\)
\(L(\frac{3}{2})\) 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 \( 1 + 7iT^{2} \)
11 \( 1 - 5.19iT - 11T^{2} \)
13 \( 1 + (3.67 - 3.67i)T - 13iT^{2} \)
17 \( 1 + (2.12 - 2.12i)T - 17iT^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
23 \( 1 + (-2.12 - 2.12i)T + 23iT^{2} \)
29 \( 1 - 5.19T + 29T^{2} \)
31 \( 1 + 5T + 31T^{2} \)
37 \( 1 + 37iT^{2} \)
41 \( 1 - 10.3iT - 41T^{2} \)
43 \( 1 + (3.67 - 3.67i)T - 43iT^{2} \)
47 \( 1 + (-6.36 + 6.36i)T - 47iT^{2} \)
53 \( 1 + (8.48 + 8.48i)T + 53iT^{2} \)
59 \( 1 - 10.3T + 59T^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 + (7.34 + 7.34i)T + 67iT^{2} \)
71 \( 1 - 10.3iT - 71T^{2} \)
73 \( 1 + (7.34 - 7.34i)T - 73iT^{2} \)
79 \( 1 - iT - 79T^{2} \)
83 \( 1 + (-8.48 - 8.48i)T + 83iT^{2} \)
89 \( 1 - 10.3T + 89T^{2} \)
97 \( 1 + (7.34 + 7.34i)T + 97iT^{2} \)
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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

−9.835120510740768999141360789824, −9.219801912709166154308376669973, −8.114830119455135498121003096942, −7.04951395711237523617101419593, −6.59174540353795646134034087915, −5.25017570557098356730258596000, −4.61578535339408099263472207263, −3.79887924173657110817517578154, −2.44641920364435099892125134425, −1.66797192772868878072795962864, 0.51566491372296849110385127954, 2.58469100551412304650657732011, 3.30346549876554805154311316790, 4.51680975496257484687321897834, 5.35518313572595516412964187413, 6.04595852756926844809352998030, 7.03094621496422206128087934768, 7.72437233711324526007705253445, 8.646603544144597664222486531015, 9.201046855341592677771410624377

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