Properties

Label 2-3234-1.1-c1-0-51
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 2.47·5-s − 6-s + 8-s + 9-s − 2.47·10-s + 11-s − 12-s + 3.03·13-s + 2.47·15-s + 16-s − 6.64·17-s + 18-s + 0.557·19-s − 2.47·20-s + 22-s − 2.66·23-s − 24-s + 1.11·25-s + 3.03·26-s − 27-s + 3.77·29-s + 2.47·30-s + 2.00·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.10·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.782·10-s + 0.301·11-s − 0.288·12-s + 0.840·13-s + 0.638·15-s + 0.250·16-s − 1.61·17-s + 0.235·18-s + 0.127·19-s − 0.553·20-s + 0.213·22-s − 0.556·23-s − 0.204·24-s + 0.223·25-s + 0.594·26-s − 0.192·27-s + 0.701·29-s + 0.451·30-s + 0.360·31-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3234,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.47T + 5T^{2} \)
13 \( 1 - 3.03T + 13T^{2} \)
17 \( 1 + 6.64T + 17T^{2} \)
19 \( 1 - 0.557T + 19T^{2} \)
23 \( 1 + 2.66T + 23T^{2} \)
29 \( 1 - 3.77T + 29T^{2} \)
31 \( 1 - 2.00T + 31T^{2} \)
37 \( 1 + 0.158T + 37T^{2} \)
41 \( 1 + 5.93T + 41T^{2} \)
43 \( 1 - 4.32T + 43T^{2} \)
47 \( 1 + 7.38T + 47T^{2} \)
53 \( 1 - 2.83T + 53T^{2} \)
59 \( 1 + 1.48T + 59T^{2} \)
61 \( 1 - 3.19T + 61T^{2} \)
67 \( 1 + 11.2T + 67T^{2} \)
71 \( 1 + 11.0T + 71T^{2} \)
73 \( 1 - 3.18T + 73T^{2} \)
79 \( 1 + 6.95T + 79T^{2} \)
83 \( 1 - 4.59T + 83T^{2} \)
89 \( 1 + 6.62T + 89T^{2} \)
97 \( 1 + 1.17T + 97T^{2} \)
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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

−8.229552755066141762483176158352, −7.36544968066836063622287394991, −6.62058980548539922510918787809, −6.11170651834801674807677798253, −5.07905832349023866780919550734, −4.27325768350918234233295067628, −3.86063421517044000959480315970, −2.76898528601165639550139358873, −1.47145706827377566419116662127, 0, 1.47145706827377566419116662127, 2.76898528601165639550139358873, 3.86063421517044000959480315970, 4.27325768350918234233295067628, 5.07905832349023866780919550734, 6.11170651834801674807677798253, 6.62058980548539922510918787809, 7.36544968066836063622287394991, 8.229552755066141762483176158352

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