Properties

Label 2-303450-1.1-c1-0-118
Degree $2$
Conductor $303450$
Sign $-1$
Analytic cond. $2423.06$
Root an. cond. $49.2245$
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 − 6-s − 7-s + 8-s + 9-s + 11-s − 12-s + 2·13-s − 14-s + 16-s + 18-s + 21-s + 22-s + 23-s − 24-s + 2·26-s − 27-s − 28-s − 6·29-s − 8·31-s + 32-s − 33-s + 36-s − 3·37-s − 2·39-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.301·11-s − 0.288·12-s + 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.235·18-s + 0.218·21-s + 0.213·22-s + 0.208·23-s − 0.204·24-s + 0.392·26-s − 0.192·27-s − 0.188·28-s − 1.11·29-s − 1.43·31-s + 0.176·32-s − 0.174·33-s + 1/6·36-s − 0.493·37-s − 0.320·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(303450\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(2423.06\)
Root analytic conductor: \(49.2245\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 303450,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 + T \)
17 \( 1 \)
good11 \( 1 - T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 + 7 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 15 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−12.94546876711597, −12.50256415207214, −12.00983663476746, −11.56290802362049, −11.13440631822783, −10.81552638193256, −10.26040104083912, −9.801086807322867, −9.236878335586016, −8.841459276210382, −8.244056788579502, −7.612246126156564, −7.196285799864097, −6.665947528485976, −6.378083554910528, −5.721964817145824, −5.324024149659517, −5.028672665109112, −4.162955059909950, −3.798785589926943, −3.457073075509556, −2.732575534160035, −1.998987388024904, −1.566872125006838, −0.7814752139623046, 0, 0.7814752139623046, 1.566872125006838, 1.998987388024904, 2.732575534160035, 3.457073075509556, 3.798785589926943, 4.162955059909950, 5.028672665109112, 5.324024149659517, 5.721964817145824, 6.378083554910528, 6.665947528485976, 7.196285799864097, 7.612246126156564, 8.244056788579502, 8.841459276210382, 9.236878335586016, 9.801086807322867, 10.26040104083912, 10.81552638193256, 11.13440631822783, 11.56290802362049, 12.00983663476746, 12.50256415207214, 12.94546876711597

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