Properties

Label 2-3234-1.1-c1-0-25
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 − 1.23·5-s + 6-s + 8-s + 9-s − 1.23·10-s − 11-s + 12-s − 1.23·15-s + 16-s + 5.23·17-s + 18-s + 7.70·19-s − 1.23·20-s − 22-s − 2.47·23-s + 24-s − 3.47·25-s + 27-s + 4.47·29-s − 1.23·30-s + 2.76·31-s + 32-s − 33-s + 5.23·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.552·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.390·10-s − 0.301·11-s + 0.288·12-s − 0.319·15-s + 0.250·16-s + 1.26·17-s + 0.235·18-s + 1.76·19-s − 0.276·20-s − 0.213·22-s − 0.515·23-s + 0.204·24-s − 0.694·25-s + 0.192·27-s + 0.830·29-s − 0.225·30-s + 0.496·31-s + 0.176·32-s − 0.174·33-s + 0.897·34-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

\(L(1)\) \(\approx\) \(3.549554618\)
\(L(\frac12)\) \(\approx\) \(3.549554618\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.23T + 5T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 5.23T + 17T^{2} \)
19 \( 1 - 7.70T + 19T^{2} \)
23 \( 1 + 2.47T + 23T^{2} \)
29 \( 1 - 4.47T + 29T^{2} \)
31 \( 1 - 2.76T + 31T^{2} \)
37 \( 1 + 10.9T + 37T^{2} \)
41 \( 1 - 5.23T + 41T^{2} \)
43 \( 1 - 6.47T + 43T^{2} \)
47 \( 1 + 3.70T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 - 1.52T + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 - 15.4T + 67T^{2} \)
71 \( 1 + 2.47T + 71T^{2} \)
73 \( 1 - 15.7T + 73T^{2} \)
79 \( 1 - 11.4T + 79T^{2} \)
83 \( 1 + 1.23T + 83T^{2} \)
89 \( 1 - 6.47T + 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−8.406468254377956189503924274561, −7.77917153838629961445706251782, −7.35301248615752742884404015787, −6.36407392004671373273687794092, −5.44482889569002572148455740427, −4.83359683909523356522985881653, −3.70116916744900971294743122738, −3.33255447285717329522932116873, −2.30622061104618036960300448251, −1.04108303344521762954040797548, 1.04108303344521762954040797548, 2.30622061104618036960300448251, 3.33255447285717329522932116873, 3.70116916744900971294743122738, 4.83359683909523356522985881653, 5.44482889569002572148455740427, 6.36407392004671373273687794092, 7.35301248615752742884404015787, 7.77917153838629961445706251782, 8.406468254377956189503924274561

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