Properties

Label 2-3234-1.1-c1-0-10
Degree $2$
Conductor $3234$
Sign $1$
Analytic cond. $25.8236$
Root an. cond. $5.08169$
Motivic weight $1$
Arithmetic yes
Rational no
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 − 2.64·5-s − 6-s − 8-s + 9-s + 2.64·10-s + 11-s + 12-s + 4·13-s − 2.64·15-s + 16-s + 3·17-s − 18-s − 5.29·19-s − 2.64·20-s − 22-s − 2.64·23-s − 24-s + 2.00·25-s − 4·26-s + 27-s + 2·29-s + 2.64·30-s + 4·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.18·5-s − 0.408·6-s − 0.353·8-s + 0.333·9-s + 0.836·10-s + 0.301·11-s + 0.288·12-s + 1.10·13-s − 0.683·15-s + 0.250·16-s + 0.727·17-s − 0.235·18-s − 1.21·19-s − 0.591·20-s − 0.213·22-s − 0.551·23-s − 0.204·24-s + 0.400·25-s − 0.784·26-s + 0.192·27-s + 0.371·29-s + 0.483·30-s + 0.718·31-s − 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3234\)    =    \(2 \cdot 3 \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(25.8236\)
Root analytic conductor: \(5.08169\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3234} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3234,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.293224027\)
\(L(\frac12)\) \(\approx\) \(1.293224027\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 - T \)
good5 \( 1 + 2.64T + 5T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 + 5.29T + 19T^{2} \)
23 \( 1 + 2.64T + 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 - 1.29T + 37T^{2} \)
41 \( 1 + 9T + 41T^{2} \)
43 \( 1 - 9.29T + 43T^{2} \)
47 \( 1 - 3.93T + 47T^{2} \)
53 \( 1 - 4T + 53T^{2} \)
59 \( 1 + 6.58T + 59T^{2} \)
61 \( 1 + 11.9T + 61T^{2} \)
67 \( 1 - 7.58T + 67T^{2} \)
71 \( 1 + 2.70T + 71T^{2} \)
73 \( 1 - 15.2T + 73T^{2} \)
79 \( 1 - 10.6T + 79T^{2} \)
83 \( 1 + 15.5T + 83T^{2} \)
89 \( 1 + 2.70T + 89T^{2} \)
97 \( 1 - 11.5T + 97T^{2} \)
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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

−8.470235918383574025210388645234, −8.119481213758977546403198644653, −7.43561891862866588715401998641, −6.59172544213212610393390989694, −5.88194655561563126909706037517, −4.50655410260769816386160914600, −3.83178771513609563236988613981, −3.10975125577073049147902862156, −1.92624059830205828812394529498, −0.74548602570690916536978104487, 0.74548602570690916536978104487, 1.92624059830205828812394529498, 3.10975125577073049147902862156, 3.83178771513609563236988613981, 4.50655410260769816386160914600, 5.88194655561563126909706037517, 6.59172544213212610393390989694, 7.43561891862866588715401998641, 8.119481213758977546403198644653, 8.470235918383574025210388645234

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