Properties

Label 2-3234-1.1-c1-0-36
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

\(L(1)\) \(\approx\) \(2.487667238\)
\(L(\frac12)\) \(\approx\) \(2.487667238\)
\(L(\frac{3}{2})\) 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 \( 1 + 9.29T + 37T^{2} \)
41 \( 1 + 9T + 41T^{2} \)
43 \( 1 + 1.29T + 43T^{2} \)
47 \( 1 + 11.9T + 47T^{2} \)
53 \( 1 - 4T + 53T^{2} \)
59 \( 1 - 14.5T + 59T^{2} \)
61 \( 1 - 3.93T + 61T^{2} \)
67 \( 1 + 13.5T + 67T^{2} \)
71 \( 1 + 13.2T + 71T^{2} \)
73 \( 1 - 4.70T + 73T^{2} \)
79 \( 1 - 5.35T + 79T^{2} \)
83 \( 1 - 5.58T + 83T^{2} \)
89 \( 1 + 13.2T + 89T^{2} \)
97 \( 1 + 9.58T + 97T^{2} \)
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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

−8.696189294109841831527476580199, −8.169971854920359980332266304665, −7.16554146366174752870743327077, −6.55660002192121550067105612549, −5.73323795139435005933552708828, −5.01660986466511968432386910326, −3.60081397328304825178492172226, −2.97226777859884400697404589723, −1.79510108368002855792555073389, −1.15446190102172941289420198828, 1.15446190102172941289420198828, 1.79510108368002855792555073389, 2.97226777859884400697404589723, 3.60081397328304825178492172226, 5.01660986466511968432386910326, 5.73323795139435005933552708828, 6.55660002192121550067105612549, 7.16554146366174752870743327077, 8.169971854920359980332266304665, 8.696189294109841831527476580199

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