Properties

Degree $2$
Conductor $25270$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 5-s − 7-s − 8-s − 3·9-s + 10-s + 4·11-s + 6·13-s + 14-s + 16-s + 2·17-s + 3·18-s − 20-s − 4·22-s + 25-s − 6·26-s − 28-s − 6·29-s − 8·31-s − 32-s − 2·34-s + 35-s − 3·36-s + 10·37-s + 40-s − 2·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s − 0.353·8-s − 9-s + 0.316·10-s + 1.20·11-s + 1.66·13-s + 0.267·14-s + 1/4·16-s + 0.485·17-s + 0.707·18-s − 0.223·20-s − 0.852·22-s + 1/5·25-s − 1.17·26-s − 0.188·28-s − 1.11·29-s − 1.43·31-s − 0.176·32-s − 0.342·34-s + 0.169·35-s − 1/2·36-s + 1.64·37-s + 0.158·40-s − 0.312·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25270\)    =    \(2 \cdot 5 \cdot 7 \cdot 19^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{25270} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 25270,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.438318010\)
\(L(\frac12)\) \(\approx\) \(1.438318010\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 + T \)
19 \( 1 \)
good3 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−15.23317622348198, −15.04166674190616, −14.28647383754939, −13.88089664724622, −13.18578504565845, −12.53812068268460, −12.00860152447218, −11.37633344545498, −11.01698575111196, −10.70267546915459, −9.625166323545191, −9.203891582574269, −8.920306414877943, −8.092394025994163, −7.838877188450487, −6.882400663204092, −6.483880069618906, −5.765250443774450, −5.434075337204800, −4.020925913496868, −3.802877685874144, −3.104366824415347, −2.207610432787885, −1.279866257568805, −0.5883617411067483, 0.5883617411067483, 1.279866257568805, 2.207610432787885, 3.104366824415347, 3.802877685874144, 4.020925913496868, 5.434075337204800, 5.765250443774450, 6.483880069618906, 6.882400663204092, 7.838877188450487, 8.092394025994163, 8.920306414877943, 9.203891582574269, 9.625166323545191, 10.70267546915459, 11.01698575111196, 11.37633344545498, 12.00860152447218, 12.53812068268460, 13.18578504565845, 13.88089664724622, 14.28647383754939, 15.04166674190616, 15.23317622348198

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