Properties

Degree $2$
Conductor $176610$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s + 5-s − 6-s + 7-s + 8-s + 9-s + 10-s − 12-s + 2·13-s + 14-s − 15-s + 16-s + 6·17-s + 18-s − 8·19-s + 20-s − 21-s − 24-s + 25-s + 2·26-s − 27-s + 28-s − 30-s + 4·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s + 0.554·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 1.83·19-s + 0.223·20-s − 0.218·21-s − 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s + 0.188·28-s − 0.182·30-s + 0.718·31-s + 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 176610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(176610\)    =    \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 29^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{176610} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 176610,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.244020341\)
\(L(\frac12)\) \(\approx\) \(5.244020341\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 - T \)
29 \( 1 \)
good11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−13.13099109383514, −12.61152874031487, −12.44648127338623, −11.79581780451425, −11.29880702521282, −10.85418385665603, −10.56725254663535, −9.942550041004715, −9.532361935576713, −8.879052757316177, −8.274205484472127, −7.779027807063295, −7.434671536464343, −6.508655586011077, −6.260054492255831, −5.952867439261224, −5.297025755079636, −4.803287552979304, −4.302711486808509, −3.816294752769613, −3.144916390278746, −2.437711955030131, −1.957424991436991, −1.167582479999241, −0.6586203238922446, 0.6586203238922446, 1.167582479999241, 1.957424991436991, 2.437711955030131, 3.144916390278746, 3.816294752769613, 4.302711486808509, 4.803287552979304, 5.297025755079636, 5.952867439261224, 6.260054492255831, 6.508655586011077, 7.434671536464343, 7.779027807063295, 8.274205484472127, 8.879052757316177, 9.532361935576713, 9.942550041004715, 10.56725254663535, 10.85418385665603, 11.29880702521282, 11.79581780451425, 12.44648127338623, 12.61152874031487, 13.13099109383514

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