Properties

Label 2-3234-1.1-c1-0-40
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 + 1.61·5-s + 6-s + 8-s + 9-s + 1.61·10-s − 11-s + 12-s + 1.61·15-s + 16-s + 7.34·17-s + 18-s − 1.17·19-s + 1.61·20-s − 22-s + 3.55·23-s + 24-s − 2.39·25-s + 27-s + 10.3·29-s + 1.61·30-s − 6.34·31-s + 32-s − 33-s + 7.34·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.721·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s + 0.510·10-s − 0.301·11-s + 0.288·12-s + 0.416·15-s + 0.250·16-s + 1.78·17-s + 0.235·18-s − 0.268·19-s + 0.360·20-s − 0.213·22-s + 0.741·23-s + 0.204·24-s − 0.479·25-s + 0.192·27-s + 1.93·29-s + 0.294·30-s − 1.13·31-s + 0.176·32-s − 0.174·33-s + 1.25·34-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

\(L(1)\) \(\approx\) \(4.473187634\)
\(L(\frac12)\) \(\approx\) \(4.473187634\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.61T + 5T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 7.34T + 17T^{2} \)
19 \( 1 + 1.17T + 19T^{2} \)
23 \( 1 - 3.55T + 23T^{2} \)
29 \( 1 - 10.3T + 29T^{2} \)
31 \( 1 + 6.34T + 31T^{2} \)
37 \( 1 - 2.82T + 37T^{2} \)
41 \( 1 + 4.94T + 41T^{2} \)
43 \( 1 + 11.5T + 43T^{2} \)
47 \( 1 + 6.78T + 47T^{2} \)
53 \( 1 - 8T + 53T^{2} \)
59 \( 1 + 4.39T + 59T^{2} \)
61 \( 1 - 11.9T + 61T^{2} \)
67 \( 1 - 2.94T + 67T^{2} \)
71 \( 1 - 15.5T + 71T^{2} \)
73 \( 1 - 5.11T + 73T^{2} \)
79 \( 1 - 5.61T + 79T^{2} \)
83 \( 1 + 5.05T + 83T^{2} \)
89 \( 1 - 0.773T + 89T^{2} \)
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

−8.455519488838681003497611920736, −7.990900185347315658325822616636, −7.01963751846613370843709631974, −6.40022878501910480091516913955, −5.40965087747041634572870630477, −5.00320645232063991520334346680, −3.79366666298471583255980512330, −3.12169860921723803499547760141, −2.24364888084564728255154563868, −1.22960168349114243527410817194, 1.22960168349114243527410817194, 2.24364888084564728255154563868, 3.12169860921723803499547760141, 3.79366666298471583255980512330, 5.00320645232063991520334346680, 5.40965087747041634572870630477, 6.40022878501910480091516913955, 7.01963751846613370843709631974, 7.990900185347315658325822616636, 8.455519488838681003497611920736

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