Properties

Label 2-175-1.1-c1-0-7
Degree $2$
Conductor $175$
Sign $1$
Analytic cond. $1.39738$
Root an. cond. $1.18210$
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  + 2·2-s + 3-s + 2·4-s + 2·6-s − 7-s − 2·9-s − 3·11-s + 2·12-s + 13-s − 2·14-s − 4·16-s + 7·17-s − 4·18-s − 21-s − 6·22-s + 6·23-s + 2·26-s − 5·27-s − 2·28-s − 5·29-s + 2·31-s − 8·32-s − 3·33-s + 14·34-s − 4·36-s + 2·37-s + 39-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.577·3-s + 4-s + 0.816·6-s − 0.377·7-s − 2/3·9-s − 0.904·11-s + 0.577·12-s + 0.277·13-s − 0.534·14-s − 16-s + 1.69·17-s − 0.942·18-s − 0.218·21-s − 1.27·22-s + 1.25·23-s + 0.392·26-s − 0.962·27-s − 0.377·28-s − 0.928·29-s + 0.359·31-s − 1.41·32-s − 0.522·33-s + 2.40·34-s − 2/3·36-s + 0.328·37-s + 0.160·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(1.39738\)
Root analytic conductor: \(1.18210\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.347442534\)
\(L(\frac12)\) \(\approx\) \(2.347442534\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 + T \)
good2 \( 1 - p T + p T^{2} \)
3 \( 1 - T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 5 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−13.01254965419346541735155009772, −12.04804768145326897305333226930, −11.05181111943764343536756650509, −9.734749354117359778149997929022, −8.569343753087953468815992872802, −7.38451703018051703490900915946, −5.93898740434763322465740513982, −5.14936161735373762545181523253, −3.57752582126466520607951640819, −2.76472422088648867371864611007, 2.76472422088648867371864611007, 3.57752582126466520607951640819, 5.14936161735373762545181523253, 5.93898740434763322465740513982, 7.38451703018051703490900915946, 8.569343753087953468815992872802, 9.734749354117359778149997929022, 11.05181111943764343536756650509, 12.04804768145326897305333226930, 13.01254965419346541735155009772

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