Properties

Label 2-2254-1.1-c3-0-170
Degree $2$
Conductor $2254$
Sign $1$
Analytic cond. $132.990$
Root an. cond. $11.5321$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 9·3-s + 4·4-s + 20·5-s + 18·6-s + 8·8-s + 54·9-s + 40·10-s − 52·11-s + 36·12-s − 43·13-s + 180·15-s + 16·16-s + 50·17-s + 108·18-s + 74·19-s + 80·20-s − 104·22-s − 23·23-s + 72·24-s + 275·25-s − 86·26-s + 243·27-s − 7·29-s + 360·30-s + 273·31-s + 32·32-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.73·3-s + 1/2·4-s + 1.78·5-s + 1.22·6-s + 0.353·8-s + 2·9-s + 1.26·10-s − 1.42·11-s + 0.866·12-s − 0.917·13-s + 3.09·15-s + 1/4·16-s + 0.713·17-s + 1.41·18-s + 0.893·19-s + 0.894·20-s − 1.00·22-s − 0.208·23-s + 0.612·24-s + 11/5·25-s − 0.648·26-s + 1.73·27-s − 0.0448·29-s + 2.19·30-s + 1.58·31-s + 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2254\)    =    \(2 \cdot 7^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(132.990\)
Root analytic conductor: \(11.5321\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2254,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(10.03157132\)
\(L(\frac12)\) \(\approx\) \(10.03157132\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p T \)
7 \( 1 \)
23 \( 1 + p T \)
good3 \( 1 - p^{2} T + p^{3} T^{2} \)
5 \( 1 - 4 p T + p^{3} T^{2} \)
11 \( 1 + 52 T + p^{3} T^{2} \)
13 \( 1 + 43 T + p^{3} T^{2} \)
17 \( 1 - 50 T + p^{3} T^{2} \)
19 \( 1 - 74 T + p^{3} T^{2} \)
29 \( 1 + 7 T + p^{3} T^{2} \)
31 \( 1 - 273 T + p^{3} T^{2} \)
37 \( 1 + 4 T + p^{3} T^{2} \)
41 \( 1 + 3 p T + p^{3} T^{2} \)
43 \( 1 + 152 T + p^{3} T^{2} \)
47 \( 1 + 75 T + p^{3} T^{2} \)
53 \( 1 - 86 T + p^{3} T^{2} \)
59 \( 1 - 444 T + p^{3} T^{2} \)
61 \( 1 + 262 T + p^{3} T^{2} \)
67 \( 1 - 764 T + p^{3} T^{2} \)
71 \( 1 + 21 T + p^{3} T^{2} \)
73 \( 1 + 681 T + p^{3} T^{2} \)
79 \( 1 - 426 T + p^{3} T^{2} \)
83 \( 1 + 902 T + p^{3} T^{2} \)
89 \( 1 - 1272 T + p^{3} T^{2} \)
97 \( 1 - 342 T + p^{3} T^{2} \)
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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

−8.659854681389875957833450273372, −7.87972928665703203077477167603, −7.24975486140213180815870721441, −6.28927987981991552499363426356, −5.30068344221281925851851628527, −4.83406513841279021135075150618, −3.45046274537069994381178339315, −2.63019472605708041710583678003, −2.34838616454164099072425878016, −1.30435290499050517761385589433, 1.30435290499050517761385589433, 2.34838616454164099072425878016, 2.63019472605708041710583678003, 3.45046274537069994381178339315, 4.83406513841279021135075150618, 5.30068344221281925851851628527, 6.28927987981991552499363426356, 7.24975486140213180815870721441, 7.87972928665703203077477167603, 8.659854681389875957833450273372

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