Properties

Label 2-327990-1.1-c1-0-39
Degree $2$
Conductor $327990$
Sign $-1$
Analytic cond. $2619.01$
Root an. cond. $51.1762$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 5-s + 6-s − 4·7-s + 8-s + 9-s + 10-s + 4·11-s + 12-s + 13-s − 4·14-s + 15-s + 16-s + 6·17-s + 18-s + 4·19-s + 20-s − 4·21-s + 4·22-s − 8·23-s + 24-s + 25-s + 26-s + 27-s − 4·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s + 0.288·12-s + 0.277·13-s − 1.06·14-s + 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s − 0.872·21-s + 0.852·22-s − 1.66·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s − 0.755·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(327990\)    =    \(2 \cdot 3 \cdot 5 \cdot 13 \cdot 29^{2}\)
Sign: $-1$
Analytic conductor: \(2619.01\)
Root analytic conductor: \(51.1762\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{327990} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 327990,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 - T \)
13 \( 1 - T \)
29 \( 1 \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−12.78270031849701, −12.55994953230907, −12.06659666146859, −11.60777870602924, −11.16287848516810, −10.32766211100919, −10.08275536098445, −9.639097520462284, −9.257772354570830, −8.935935696661124, −8.128948835137126, −7.604707178607630, −7.292678014077952, −6.684845677154928, −6.109790544640027, −5.931819908988691, −5.514732374953802, −4.663908347091673, −4.074635728361258, −3.652952578597312, −3.283487545400247, −2.861872048761030, −2.154053341321339, −1.488856806822536, −1.024210034184673, 0, 1.024210034184673, 1.488856806822536, 2.154053341321339, 2.861872048761030, 3.283487545400247, 3.652952578597312, 4.074635728361258, 4.663908347091673, 5.514732374953802, 5.931819908988691, 6.109790544640027, 6.684845677154928, 7.292678014077952, 7.604707178607630, 8.128948835137126, 8.935935696661124, 9.257772354570830, 9.639097520462284, 10.08275536098445, 10.32766211100919, 11.16287848516810, 11.60777870602924, 12.06659666146859, 12.55994953230907, 12.78270031849701

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