Properties

Label 2-3234-77.76-c1-0-65
Degree $2$
Conductor $3234$
Sign $0.481 + 0.876i$
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 − 0.837i·5-s + 6-s i·8-s − 9-s + 0.837·10-s + (0.697 − 3.24i)11-s + i·12-s + 2.59·13-s − 0.837·15-s + 16-s + 5.97·17-s i·18-s − 3.11·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.374i·5-s + 0.408·6-s − 0.353i·8-s − 0.333·9-s + 0.264·10-s + (0.210 − 0.977i)11-s + 0.288i·12-s + 0.719·13-s − 0.216·15-s + 0.250·16-s + 1.44·17-s − 0.235i·18-s − 0.713·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.481 + 0.876i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.481 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3234\)    =    \(2 \cdot 3 \cdot 7^{2} \cdot 11\)
Sign: $0.481 + 0.876i$
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: \((2,\ 3234,\ (\ :1/2),\ 0.481 + 0.876i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.647336936\)
\(L(\frac12)\) \(\approx\) \(1.647336936\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.697 + 3.24i)T \)
good5 \( 1 + 0.837iT - 5T^{2} \)
13 \( 1 - 2.59T + 13T^{2} \)
17 \( 1 - 5.97T + 17T^{2} \)
19 \( 1 + 3.11T + 19T^{2} \)
23 \( 1 - 2.86T + 23T^{2} \)
29 \( 1 + 5.38iT - 29T^{2} \)
31 \( 1 + 1.05iT - 31T^{2} \)
37 \( 1 + 10.9T + 37T^{2} \)
41 \( 1 + 11.2T + 41T^{2} \)
43 \( 1 - 1.27iT - 43T^{2} \)
47 \( 1 - 12.2iT - 47T^{2} \)
53 \( 1 - 5.17T + 53T^{2} \)
59 \( 1 + 9.68iT - 59T^{2} \)
61 \( 1 - 4.07T + 61T^{2} \)
67 \( 1 - 13.0T + 67T^{2} \)
71 \( 1 - 14.0T + 71T^{2} \)
73 \( 1 + 9.91T + 73T^{2} \)
79 \( 1 + 13.5iT - 79T^{2} \)
83 \( 1 - 2.99T + 83T^{2} \)
89 \( 1 + 8.40iT - 89T^{2} \)
97 \( 1 + 0.786iT - 97T^{2} \)
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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

−8.406892063220956125952575493527, −7.909050230118605014722963850852, −6.93094642245553641262496715107, −6.34135104571286379615527398192, −5.61181039317166959073881037406, −4.95493211923927642448926652164, −3.77866400901678804919669614487, −3.07244794951625379134180191500, −1.55988140387697961364971953728, −0.58033367246852718681148359998, 1.20562731012024759617247272530, 2.27849741896284531121492285180, 3.42547817296306050441622592102, 3.76774398568307565958347655258, 5.00619521796640476110546494231, 5.36214407811775396897231675461, 6.67162248085925426698089729630, 7.16923946605742512343508044686, 8.443108568482292423076016396217, 8.738737037072673697653912318401

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