Properties

Label 2-175-1.1-c3-0-11
Degree $2$
Conductor $175$
Sign $1$
Analytic cond. $10.3253$
Root an. cond. $3.21330$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.84·2-s + 9.58·3-s + 15.4·4-s − 46.3·6-s − 7·7-s − 36.0·8-s + 64.7·9-s + 62.1·11-s + 147.·12-s − 14.0·13-s + 33.8·14-s + 50.9·16-s − 63.5·17-s − 313.·18-s + 48.7·19-s − 67.0·21-s − 301.·22-s + 99.3·23-s − 345.·24-s + 68.2·26-s + 362.·27-s − 108.·28-s − 69.0·29-s − 9.68·31-s + 41.7·32-s + 595.·33-s + 307.·34-s + ⋯
L(s)  = 1  − 1.71·2-s + 1.84·3-s + 1.93·4-s − 3.15·6-s − 0.377·7-s − 1.59·8-s + 2.39·9-s + 1.70·11-s + 3.55·12-s − 0.300·13-s + 0.646·14-s + 0.795·16-s − 0.906·17-s − 4.10·18-s + 0.588·19-s − 0.696·21-s − 2.91·22-s + 0.900·23-s − 2.93·24-s + 0.514·26-s + 2.58·27-s − 0.729·28-s − 0.442·29-s − 0.0561·31-s + 0.230·32-s + 3.14·33-s + 1.55·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(10.3253\)
Root analytic conductor: \(3.21330\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.544092944\)
\(L(\frac12)\) \(\approx\) \(1.544092944\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 + 7T \)
good2 \( 1 + 4.84T + 8T^{2} \)
3 \( 1 - 9.58T + 27T^{2} \)
11 \( 1 - 62.1T + 1.33e3T^{2} \)
13 \( 1 + 14.0T + 2.19e3T^{2} \)
17 \( 1 + 63.5T + 4.91e3T^{2} \)
19 \( 1 - 48.7T + 6.85e3T^{2} \)
23 \( 1 - 99.3T + 1.21e4T^{2} \)
29 \( 1 + 69.0T + 2.43e4T^{2} \)
31 \( 1 + 9.68T + 2.97e4T^{2} \)
37 \( 1 + 240.T + 5.06e4T^{2} \)
41 \( 1 - 335.T + 6.89e4T^{2} \)
43 \( 1 + 51.2T + 7.95e4T^{2} \)
47 \( 1 - 451.T + 1.03e5T^{2} \)
53 \( 1 - 180.T + 1.48e5T^{2} \)
59 \( 1 - 268.T + 2.05e5T^{2} \)
61 \( 1 + 323.T + 2.26e5T^{2} \)
67 \( 1 - 541.T + 3.00e5T^{2} \)
71 \( 1 + 161.T + 3.57e5T^{2} \)
73 \( 1 + 305.T + 3.89e5T^{2} \)
79 \( 1 + 504.T + 4.93e5T^{2} \)
83 \( 1 - 513.T + 5.71e5T^{2} \)
89 \( 1 - 543.T + 7.04e5T^{2} \)
97 \( 1 + 1.86e3T + 9.12e5T^{2} \)
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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

−12.09283125001758694192565456948, −10.80223475511244050995761026133, −9.592529723979436201608869094205, −9.183737465872716343535020810613, −8.556627141196053326647380862039, −7.37911752003919553288537713499, −6.75822527144168371605956012013, −3.94033848476198873690756859061, −2.57098019204571101580261907459, −1.31829429226928772840387860829, 1.31829429226928772840387860829, 2.57098019204571101580261907459, 3.94033848476198873690756859061, 6.75822527144168371605956012013, 7.37911752003919553288537713499, 8.556627141196053326647380862039, 9.183737465872716343535020810613, 9.592529723979436201608869094205, 10.80223475511244050995761026133, 12.09283125001758694192565456948

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