Properties

Label 2-3234-1.1-c1-0-7
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 − 2.80·5-s − 6-s + 8-s + 9-s − 2.80·10-s − 11-s − 12-s + 2.80·15-s + 16-s + 7.96·17-s + 18-s − 6.48·19-s − 2.80·20-s − 22-s − 5.28·23-s − 24-s + 2.87·25-s − 27-s + 5.12·29-s + 2.80·30-s − 8.96·31-s + 32-s + 33-s + 7.96·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.25·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.887·10-s − 0.301·11-s − 0.288·12-s + 0.724·15-s + 0.250·16-s + 1.93·17-s + 0.235·18-s − 1.48·19-s − 0.627·20-s − 0.213·22-s − 1.10·23-s − 0.204·24-s + 0.574·25-s − 0.192·27-s + 0.952·29-s + 0.512·30-s − 1.61·31-s + 0.176·32-s + 0.174·33-s + 1.36·34-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

\(L(1)\) \(\approx\) \(1.637133907\)
\(L(\frac12)\) \(\approx\) \(1.637133907\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.80T + 5T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 7.96T + 17T^{2} \)
19 \( 1 + 6.48T + 19T^{2} \)
23 \( 1 + 5.28T + 23T^{2} \)
29 \( 1 - 5.12T + 29T^{2} \)
31 \( 1 + 8.96T + 31T^{2} \)
37 \( 1 - 10.4T + 37T^{2} \)
41 \( 1 + 5.09T + 41T^{2} \)
43 \( 1 - 11.4T + 43T^{2} \)
47 \( 1 - 0.322T + 47T^{2} \)
53 \( 1 - 8T + 53T^{2} \)
59 \( 1 + 0.871T + 59T^{2} \)
61 \( 1 - 2.15T + 61T^{2} \)
67 \( 1 + 7.09T + 67T^{2} \)
71 \( 1 + 7.44T + 71T^{2} \)
73 \( 1 - 12.5T + 73T^{2} \)
79 \( 1 - 6.80T + 79T^{2} \)
83 \( 1 - 15.0T + 83T^{2} \)
89 \( 1 - 1.61T + 89T^{2} \)
97 \( 1 - 7T + 97T^{2} \)
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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

−8.322893067985681849778390099941, −7.77355276847492292262559649407, −7.22879310715581657192938050079, −6.18265288791516062719991365611, −5.66638775860792254168616195747, −4.68664464047049222632433931200, −4.02297614110335763700332428568, −3.38250537199491794706764140625, −2.17400576853987307769344817120, −0.70023435125508330428256770482, 0.70023435125508330428256770482, 2.17400576853987307769344817120, 3.38250537199491794706764140625, 4.02297614110335763700332428568, 4.68664464047049222632433931200, 5.66638775860792254168616195747, 6.18265288791516062719991365611, 7.22879310715581657192938050079, 7.77355276847492292262559649407, 8.322893067985681849778390099941

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