Properties

Label 2-29645-1.1-c1-0-9
Degree $2$
Conductor $29645$
Sign $-1$
Analytic cond. $236.716$
Root an. cond. $15.3855$
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 + 3·3-s − 4-s − 5-s − 3·6-s + 3·8-s + 6·9-s + 10-s − 3·12-s − 4·13-s − 3·15-s − 16-s − 6·18-s − 4·19-s + 20-s − 8·23-s + 9·24-s + 25-s + 4·26-s + 9·27-s + 6·29-s + 3·30-s + 2·31-s − 5·32-s − 6·36-s − 8·37-s + 4·38-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.73·3-s − 1/2·4-s − 0.447·5-s − 1.22·6-s + 1.06·8-s + 2·9-s + 0.316·10-s − 0.866·12-s − 1.10·13-s − 0.774·15-s − 1/4·16-s − 1.41·18-s − 0.917·19-s + 0.223·20-s − 1.66·23-s + 1.83·24-s + 1/5·25-s + 0.784·26-s + 1.73·27-s + 1.11·29-s + 0.547·30-s + 0.359·31-s − 0.883·32-s − 36-s − 1.31·37-s + 0.648·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29645 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29645\)    =    \(5 \cdot 7^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(236.716\)
Root analytic conductor: \(15.3855\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{29645} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 29645,\ (\ :1/2),\ -1)\)

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)$
bad5 \( 1 + T \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + T + p T^{2} \)
3 \( 1 - p T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 - 11 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + T + p T^{2} \)
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

−15.34326032447909, −14.75457709466858, −14.38283872587844, −13.84955010337413, −13.58953017393668, −12.75155638617457, −12.39640225690533, −11.84225616449429, −10.79587244392930, −10.27818806715555, −9.902274231445175, −9.377420094232372, −8.778014970433280, −8.368786011913590, −8.010307359304773, −7.399540731593491, −7.014810615697225, −6.082830294727432, −5.049305042470314, −4.410548178020603, −4.002614774195866, −3.365307367226744, −2.373630893777074, −2.120414393881849, −1.038664888659995, 0, 1.038664888659995, 2.120414393881849, 2.373630893777074, 3.365307367226744, 4.002614774195866, 4.410548178020603, 5.049305042470314, 6.082830294727432, 7.014810615697225, 7.399540731593491, 8.010307359304773, 8.368786011913590, 8.778014970433280, 9.377420094232372, 9.902274231445175, 10.27818806715555, 10.79587244392930, 11.84225616449429, 12.39640225690533, 12.75155638617457, 13.58953017393668, 13.84955010337413, 14.38283872587844, 14.75457709466858, 15.34326032447909

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