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Properties

Label 2-227430-1.1-c1-0-112
Degree $2$
Conductor $227430$
Sign $-1$
Analytic cond. $1816.03$
Root an. cond. $42.6149$
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 + 4-s − 5-s + 7-s − 8-s + 10-s − 2·11-s + 4·13-s − 14-s + 16-s + 4·17-s − 20-s + 2·22-s + 25-s − 4·26-s + 28-s + 2·29-s − 10·31-s − 32-s − 4·34-s − 35-s + 12·37-s + 40-s + 2·41-s − 4·43-s − 2·44-s + 49-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s − 0.353·8-s + 0.316·10-s − 0.603·11-s + 1.10·13-s − 0.267·14-s + 1/4·16-s + 0.970·17-s − 0.223·20-s + 0.426·22-s + 1/5·25-s − 0.784·26-s + 0.188·28-s + 0.371·29-s − 1.79·31-s − 0.176·32-s − 0.685·34-s − 0.169·35-s + 1.97·37-s + 0.158·40-s + 0.312·41-s − 0.609·43-s − 0.301·44-s + 1/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 227430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 227430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(227430\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(1816.03\)
Root analytic conductor: \(42.6149\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 227430,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 + T \)
7 \( 1 - T \)
19 \( 1 \)
good11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 12 T + p T^{2} \)
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 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.00178854715768, −12.73989507782608, −12.20860477768978, −11.52955751143100, −11.27880212690706, −10.93320699037335, −10.32156598547716, −10.01353821323182, −9.303340615451655, −8.998479488060833, −8.359829001278828, −8.053401946848145, −7.504670417434946, −7.320554286956452, −6.478508695213273, −5.992312094671001, −5.630242039512290, −4.946366195991285, −4.398142338216028, −3.748719316262815, −3.238130720985792, −2.751048520776151, −1.944312860903839, −1.383223207175770, −0.8014975570641144, 0, 0.8014975570641144, 1.383223207175770, 1.944312860903839, 2.751048520776151, 3.238130720985792, 3.748719316262815, 4.398142338216028, 4.946366195991285, 5.630242039512290, 5.992312094671001, 6.478508695213273, 7.320554286956452, 7.504670417434946, 8.053401946848145, 8.359829001278828, 8.998479488060833, 9.303340615451655, 10.01353821323182, 10.32156598547716, 10.93320699037335, 11.27880212690706, 11.52955751143100, 12.20860477768978, 12.73989507782608, 13.00178854715768

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