Properties

Label 2-348726-1.1-c1-0-59
Degree $2$
Conductor $348726$
Sign $1$
Analytic cond. $2784.59$
Root an. cond. $52.7692$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 3·5-s − 6-s + 7-s − 8-s + 9-s + 3·10-s − 5·11-s + 12-s − 6·13-s − 14-s − 3·15-s + 16-s − 3·17-s − 18-s − 3·20-s + 21-s + 5·22-s + 23-s − 24-s + 4·25-s + 6·26-s + 27-s + 28-s − 6·29-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.948·10-s − 1.50·11-s + 0.288·12-s − 1.66·13-s − 0.267·14-s − 0.774·15-s + 1/4·16-s − 0.727·17-s − 0.235·18-s − 0.670·20-s + 0.218·21-s + 1.06·22-s + 0.208·23-s − 0.204·24-s + 4/5·25-s + 1.17·26-s + 0.192·27-s + 0.188·28-s − 1.11·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 348726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(348726\)    =    \(2 \cdot 3 \cdot 7 \cdot 19^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(2784.59\)
Root analytic conductor: \(52.7692\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 348726,\ (\ :1/2),\ 1)\)

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 \( 1 - T \)
7 \( 1 - T \)
19 \( 1 \)
23 \( 1 - T \)
good5 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + 11 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
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

−12.83294215948477, −12.61803586696032, −11.93292912657673, −11.68181895197847, −11.11867257961878, −10.67496940658677, −10.35882556273203, −9.797472729770305, −9.293889535319796, −8.715817949617604, −8.555053873043643, −7.765828117341412, −7.559959975577100, −7.310426605159770, −6.994194894571889, −6.017452657904012, −5.438758104932791, −4.987037100648014, −4.469748552985514, −3.929551028687089, −3.399263756648084, −2.726823115422709, −2.312951683370571, −1.929823708663729, −0.9080481332024711, 0, 0, 0.9080481332024711, 1.929823708663729, 2.312951683370571, 2.726823115422709, 3.399263756648084, 3.929551028687089, 4.469748552985514, 4.987037100648014, 5.438758104932791, 6.017452657904012, 6.994194894571889, 7.310426605159770, 7.559959975577100, 7.765828117341412, 8.555053873043643, 8.715817949617604, 9.293889535319796, 9.797472729770305, 10.35882556273203, 10.67496940658677, 11.11867257961878, 11.68181895197847, 11.93292912657673, 12.61803586696032, 12.83294215948477

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