Properties

Label 2-3234-1.1-c1-0-29
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 + 2·5-s − 6-s + 8-s + 9-s + 2·10-s − 11-s − 12-s + 5.41·13-s − 2·15-s + 16-s + 2·17-s + 18-s − 2.24·19-s + 2·20-s − 22-s − 4.82·23-s − 24-s − 25-s + 5.41·26-s − 27-s + 9.65·29-s − 2·30-s + 6.24·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.894·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s + 0.632·10-s − 0.301·11-s − 0.288·12-s + 1.50·13-s − 0.516·15-s + 0.250·16-s + 0.485·17-s + 0.235·18-s − 0.514·19-s + 0.447·20-s − 0.213·22-s − 1.00·23-s − 0.204·24-s − 0.200·25-s + 1.06·26-s − 0.192·27-s + 1.79·29-s − 0.365·30-s + 1.12·31-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: $\chi_{3234} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3234,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.164966758\)
\(L(\frac12)\) \(\approx\) \(3.164966758\)
\(L(\frac{3}{2})\) 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 \( 1 - 2T + 5T^{2} \)
13 \( 1 - 5.41T + 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + 2.24T + 19T^{2} \)
23 \( 1 + 4.82T + 23T^{2} \)
29 \( 1 - 9.65T + 29T^{2} \)
31 \( 1 - 6.24T + 31T^{2} \)
37 \( 1 - 0.828T + 37T^{2} \)
41 \( 1 - 11.6T + 41T^{2} \)
43 \( 1 + 4.82T + 43T^{2} \)
47 \( 1 + 7.89T + 47T^{2} \)
53 \( 1 + 10.4T + 53T^{2} \)
59 \( 1 - 6.82T + 59T^{2} \)
61 \( 1 - 2.58T + 61T^{2} \)
67 \( 1 + 1.17T + 67T^{2} \)
71 \( 1 - 5.65T + 71T^{2} \)
73 \( 1 - 11.6T + 73T^{2} \)
79 \( 1 + 16.4T + 79T^{2} \)
83 \( 1 - 5.07T + 83T^{2} \)
89 \( 1 - 3.75T + 89T^{2} \)
97 \( 1 + 9.89T + 97T^{2} \)
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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

−8.452700009455326605632775384571, −7.981469552544518552284372504302, −6.73738043986355651898002442427, −6.17550687159741192834472047394, −5.80831539177917704294157620685, −4.85853113041344073102923490752, −4.11552285298945100391736118888, −3.11308842732284167142129486182, −2.07382726814357683375991202659, −1.05504638080848840359357740156, 1.05504638080848840359357740156, 2.07382726814357683375991202659, 3.11308842732284167142129486182, 4.11552285298945100391736118888, 4.85853113041344073102923490752, 5.80831539177917704294157620685, 6.17550687159741192834472047394, 6.73738043986355651898002442427, 7.981469552544518552284372504302, 8.452700009455326605632775384571

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