Properties

Label 2-350-1.1-c3-0-25
Degree $2$
Conductor $350$
Sign $-1$
Analytic cond. $20.6506$
Root an. cond. $4.54430$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 7·3-s + 4·4-s − 14·6-s − 7·7-s − 8·8-s + 22·9-s − 37·11-s + 28·12-s − 51·13-s + 14·14-s + 16·16-s − 41·17-s − 44·18-s − 108·19-s − 49·21-s + 74·22-s + 70·23-s − 56·24-s + 102·26-s − 35·27-s − 28·28-s − 249·29-s − 134·31-s − 32·32-s − 259·33-s + 82·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.34·3-s + 1/2·4-s − 0.952·6-s − 0.377·7-s − 0.353·8-s + 0.814·9-s − 1.01·11-s + 0.673·12-s − 1.08·13-s + 0.267·14-s + 1/4·16-s − 0.584·17-s − 0.576·18-s − 1.30·19-s − 0.509·21-s + 0.717·22-s + 0.634·23-s − 0.476·24-s + 0.769·26-s − 0.249·27-s − 0.188·28-s − 1.59·29-s − 0.776·31-s − 0.176·32-s − 1.36·33-s + 0.413·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $-1$
Analytic conductor: \(20.6506\)
Root analytic conductor: \(4.54430\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 350,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p T \)
5 \( 1 \)
7 \( 1 + p T \)
good3 \( 1 - 7 T + p^{3} T^{2} \)
11 \( 1 + 37 T + p^{3} T^{2} \)
13 \( 1 + 51 T + p^{3} T^{2} \)
17 \( 1 + 41 T + p^{3} T^{2} \)
19 \( 1 + 108 T + p^{3} T^{2} \)
23 \( 1 - 70 T + p^{3} T^{2} \)
29 \( 1 + 249 T + p^{3} T^{2} \)
31 \( 1 + 134 T + p^{3} T^{2} \)
37 \( 1 - 334 T + p^{3} T^{2} \)
41 \( 1 - 206 T + p^{3} T^{2} \)
43 \( 1 - 376 T + p^{3} T^{2} \)
47 \( 1 - 287 T + p^{3} T^{2} \)
53 \( 1 - 6 T + p^{3} T^{2} \)
59 \( 1 + 2 T + p^{3} T^{2} \)
61 \( 1 + 940 T + p^{3} T^{2} \)
67 \( 1 + 106 T + p^{3} T^{2} \)
71 \( 1 - 456 T + p^{3} T^{2} \)
73 \( 1 + 650 T + p^{3} T^{2} \)
79 \( 1 + 1239 T + p^{3} T^{2} \)
83 \( 1 + 428 T + p^{3} T^{2} \)
89 \( 1 + 220 T + p^{3} T^{2} \)
97 \( 1 - 1055 T + p^{3} T^{2} \)
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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

−10.39214305996520387906339888869, −9.353799463143717572254419749900, −8.917233041536028753055887601783, −7.77195361751981983908474916846, −7.30423702925814586430420233260, −5.84980864057896464502833975805, −4.27224722524168557518000785924, −2.84053479250650156946967116642, −2.15604093861471443380110754259, 0, 2.15604093861471443380110754259, 2.84053479250650156946967116642, 4.27224722524168557518000785924, 5.84980864057896464502833975805, 7.30423702925814586430420233260, 7.77195361751981983908474916846, 8.917233041536028753055887601783, 9.353799463143717572254419749900, 10.39214305996520387906339888869

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