Properties

Label 2-3234-77.76-c1-0-79
Degree $2$
Conductor $3234$
Sign $0.614 - 0.789i$
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 − 3.51i·5-s − 6-s + i·8-s − 9-s − 3.51·10-s + (−2.26 − 2.42i)11-s + i·12-s − 5.40·13-s − 3.51·15-s + 16-s − 2.00·17-s + i·18-s − 4.64·19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 1.57i·5-s − 0.408·6-s + 0.353i·8-s − 0.333·9-s − 1.11·10-s + (−0.683 − 0.730i)11-s + 0.288i·12-s − 1.50·13-s − 0.907·15-s + 0.250·16-s − 0.485·17-s + 0.235i·18-s − 1.06·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.614 - 0.789i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.614 - 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3234\)    =    \(2 \cdot 3 \cdot 7^{2} \cdot 11\)
Sign: $0.614 - 0.789i$
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: \((2,\ 3234,\ (\ :1/2),\ 0.614 - 0.789i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4149591223\)
\(L(\frac12)\) \(\approx\) \(0.4149591223\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.26 + 2.42i)T \)
good5 \( 1 + 3.51iT - 5T^{2} \)
13 \( 1 + 5.40T + 13T^{2} \)
17 \( 1 + 2.00T + 17T^{2} \)
19 \( 1 + 4.64T + 19T^{2} \)
23 \( 1 - 7.74T + 23T^{2} \)
29 \( 1 + 8.62iT - 29T^{2} \)
31 \( 1 + 5.01iT - 31T^{2} \)
37 \( 1 - 5.41T + 37T^{2} \)
41 \( 1 + 0.994T + 41T^{2} \)
43 \( 1 - 9.82iT - 43T^{2} \)
47 \( 1 - 10.6iT - 47T^{2} \)
53 \( 1 - 2.47T + 53T^{2} \)
59 \( 1 - 0.0847iT - 59T^{2} \)
61 \( 1 - 1.37T + 61T^{2} \)
67 \( 1 + 11.8T + 67T^{2} \)
71 \( 1 - 12.6T + 71T^{2} \)
73 \( 1 - 7.10T + 73T^{2} \)
79 \( 1 + 8.52iT - 79T^{2} \)
83 \( 1 + 3.88T + 83T^{2} \)
89 \( 1 + 3.47iT - 89T^{2} \)
97 \( 1 + 8.90iT - 97T^{2} \)
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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

−8.041346055206757617484660350310, −7.64852346938663848507721121416, −6.37705550719897375334762062625, −5.57653527908555732043570131001, −4.72957228182287485335031333088, −4.34857203872146650586167699811, −2.86229544066144882240005778682, −2.19633026747684241622918382739, −0.980800810533718307359948797383, −0.14489624252619195506748352085, 2.25396647298749869914124850719, 2.92249357767559940546631746784, 3.89512105608812690396973332885, 4.93455378015691922685095353702, 5.32830481073187308448667190423, 6.69081933539253264151043224557, 6.91166294658034876062596615961, 7.52950822888332950414671896884, 8.532456248552779955135937117792, 9.275047766448444853000116021954

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