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Properties

Label 2-74970-1.1-c1-0-38
Degree $2$
Conductor $74970$
Sign $1$
Analytic cond. $598.638$
Root an. cond. $24.4670$
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 + 4-s − 5-s − 8-s + 10-s + 4·11-s + 6·13-s + 16-s + 17-s + 4·19-s − 20-s − 4·22-s + 4·23-s + 25-s − 6·26-s − 2·29-s + 4·31-s − 32-s − 34-s − 6·37-s − 4·38-s + 40-s − 2·41-s + 4·44-s − 4·46-s + 8·47-s − 50-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s + 1.20·11-s + 1.66·13-s + 1/4·16-s + 0.242·17-s + 0.917·19-s − 0.223·20-s − 0.852·22-s + 0.834·23-s + 1/5·25-s − 1.17·26-s − 0.371·29-s + 0.718·31-s − 0.176·32-s − 0.171·34-s − 0.986·37-s − 0.648·38-s + 0.158·40-s − 0.312·41-s + 0.603·44-s − 0.589·46-s + 1.16·47-s − 0.141·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74970 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74970 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74970\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 17\)
Sign: $1$
Analytic conductor: \(598.638\)
Root analytic conductor: \(24.4670\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 74970,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.397693058\)
\(L(\frac12)\) \(\approx\) \(2.397693058\)
\(L(\frac{3}{2})\) 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 \( 1 \)
17 \( 1 - T \)
good11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
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

−14.07028127115491, −13.60248021286473, −13.12756935745035, −12.37021191067974, −11.87755792042126, −11.62104227952579, −10.96501377238621, −10.68016409198135, −9.990345286085748, −9.411752922212691, −8.914060423392801, −8.582417091403711, −8.084740950884439, −7.316071815886476, −6.991451006057924, −6.412061102505369, −5.831568071872778, −5.316521744028665, −4.431564828775510, −3.806534574865974, −3.410108331776092, −2.759783799523991, −1.733486936575561, −1.175715056490352, −0.6710944629023416, 0.6710944629023416, 1.175715056490352, 1.733486936575561, 2.759783799523991, 3.410108331776092, 3.806534574865974, 4.431564828775510, 5.316521744028665, 5.831568071872778, 6.412061102505369, 6.991451006057924, 7.316071815886476, 8.084740950884439, 8.582417091403711, 8.914060423392801, 9.411752922212691, 9.990345286085748, 10.68016409198135, 10.96501377238621, 11.62104227952579, 11.87755792042126, 12.37021191067974, 13.12756935745035, 13.60248021286473, 14.07028127115491

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