Properties

Label 2-3234-1.1-c1-0-41
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

\(L(1)\) \(\approx\) \(4.684799123\)
\(L(\frac12)\) \(\approx\) \(4.684799123\)
\(L(\frac{3}{2})\) 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

−8.708043564663346712492101037959, −7.84467749976348884436416024841, −7.05960484512122367580274885733, −6.17172279599073096756715180624, −5.77863900607316860031675382520, −4.66242801711219815870275478686, −4.14477436934902933626891039353, −2.64117314516565810080346788885, −2.53341879180570981254080568105, −1.24953794370594853295520798746, 1.24953794370594853295520798746, 2.53341879180570981254080568105, 2.64117314516565810080346788885, 4.14477436934902933626891039353, 4.66242801711219815870275478686, 5.77863900607316860031675382520, 6.17172279599073096756715180624, 7.05960484512122367580274885733, 7.84467749976348884436416024841, 8.708043564663346712492101037959

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