Properties

Label 2-3234-77.76-c1-0-51
Degree $2$
Conductor $3234$
Sign $0.613 + 0.789i$
Analytic cond. $25.8236$
Root an. cond. $5.08169$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·2-s + i·3-s − 4-s + 0.700i·5-s + 6-s + i·8-s − 9-s + 0.700·10-s + (0.176 − 3.31i)11-s i·12-s + 7.03·13-s − 0.700·15-s + 16-s − 0.617·17-s + i·18-s + 0.783·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.313i·5-s + 0.408·6-s + 0.353i·8-s − 0.333·9-s + 0.221·10-s + (0.0531 − 0.998i)11-s − 0.288i·12-s + 1.95·13-s − 0.180·15-s + 0.250·16-s − 0.149·17-s + 0.235i·18-s + 0.179·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.613 + 0.789i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.613 + 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3234\)    =    \(2 \cdot 3 \cdot 7^{2} \cdot 11\)
Sign: $0.613 + 0.789i$
Analytic conductor: \(25.8236\)
Root analytic conductor: \(5.08169\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3234} (2155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3234,\ (\ :1/2),\ 0.613 + 0.789i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.779073098\)
\(L(\frac12)\) \(\approx\) \(1.779073098\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 - iT \)
7 \( 1 \)
11 \( 1 + (-0.176 + 3.31i)T \)
good5 \( 1 - 0.700iT - 5T^{2} \)
13 \( 1 - 7.03T + 13T^{2} \)
17 \( 1 + 0.617T + 17T^{2} \)
19 \( 1 - 0.783T + 19T^{2} \)
23 \( 1 + 5.27T + 23T^{2} \)
29 \( 1 - 2.09iT - 29T^{2} \)
31 \( 1 + 6.17iT - 31T^{2} \)
37 \( 1 + 3.99T + 37T^{2} \)
41 \( 1 + 5.85T + 41T^{2} \)
43 \( 1 + 3.62iT - 43T^{2} \)
47 \( 1 - 2.34iT - 47T^{2} \)
53 \( 1 - 7.90T + 53T^{2} \)
59 \( 1 + 3.76iT - 59T^{2} \)
61 \( 1 - 8.36T + 61T^{2} \)
67 \( 1 - 2.14T + 67T^{2} \)
71 \( 1 + 4.48T + 71T^{2} \)
73 \( 1 - 5.28T + 73T^{2} \)
79 \( 1 - 4.33iT - 79T^{2} \)
83 \( 1 + 4.03T + 83T^{2} \)
89 \( 1 + 16.2iT - 89T^{2} \)
97 \( 1 + 10.4iT - 97T^{2} \)
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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

−8.532648665511387187799311259157, −8.290731466798174644317365463079, −6.98338111762034046390017069005, −6.04008886088943761924750412508, −5.57370000651727650905891528666, −4.41939413997056345282318803410, −3.62227094535407060250383067669, −3.18781653122818637208027381549, −1.93324959030994124005688238255, −0.69566157134856679793799980834, 0.997873949149393979476982268714, 1.95505442284027865956862735112, 3.36392219759847027458382498948, 4.17804527268623275404400905797, 5.10268695699761000459490054760, 5.87331701226785854637344656283, 6.63237385734768767846228898774, 7.11175338497711037635316622491, 8.135724590849270128230460217691, 8.518199780092247754833169562558

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