Properties

Label 2-327990-1.1-c1-0-12
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 5-s − 6-s + 2·7-s − 8-s + 9-s + 10-s + 12-s + 13-s − 2·14-s − 15-s + 16-s − 6·17-s − 18-s + 2·19-s − 20-s + 2·21-s + 3·23-s − 24-s + 25-s − 26-s + 27-s + 2·28-s + 30-s + 5·31-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s + 0.277·13-s − 0.534·14-s − 0.258·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.458·19-s − 0.223·20-s + 0.436·21-s + 0.625·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s + 0.377·28-s + 0.182·30-s + 0.898·31-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: \(0\)
Selberg data: \((2,\ 327990,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.520521907\)
\(L(\frac12)\) \(\approx\) \(2.520521907\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−12.64764997700626, −12.01433885334237, −11.40267823585213, −11.32798149853390, −10.89858649781760, −10.23138756911491, −9.833733871004233, −9.321736901295883, −8.899313052653279, −8.364355421527973, −8.088561716740419, −7.820247962897629, −6.985756554173421, −6.717665552051117, −6.387422694388742, −5.441043088736590, −5.002244654921933, −4.520940686196958, −3.983420430316928, −3.298254327723849, −2.945037042623153, −2.070346121628825, −1.900707567127676, −1.030004567659519, −0.5028493006475713, 0.5028493006475713, 1.030004567659519, 1.900707567127676, 2.070346121628825, 2.945037042623153, 3.298254327723849, 3.983420430316928, 4.520940686196958, 5.002244654921933, 5.441043088736590, 6.387422694388742, 6.717665552051117, 6.985756554173421, 7.820247962897629, 8.088561716740419, 8.364355421527973, 8.899313052653279, 9.321736901295883, 9.833733871004233, 10.23138756911491, 10.89858649781760, 11.32798149853390, 11.40267823585213, 12.01433885334237, 12.64764997700626

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