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Properties

Label 2-3234-77.76-c1-0-69
Degree $2$
Conductor $3234$
Sign $0.771 + 0.636i$
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 + 2.84i·5-s − 6-s − i·8-s − 9-s − 2.84·10-s + (2.48 − 2.19i)11-s − i·12-s + 0.726·13-s − 2.84·15-s + 16-s − 5.60·17-s − i·18-s − 6.09·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 1.27i·5-s − 0.408·6-s − 0.353i·8-s − 0.333·9-s − 0.899·10-s + (0.749 − 0.662i)11-s − 0.288i·12-s + 0.201·13-s − 0.734·15-s + 0.250·16-s − 1.35·17-s − 0.235i·18-s − 1.39·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.771 + 0.636i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.771 + 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3234\)    =    \(2 \cdot 3 \cdot 7^{2} \cdot 11\)
Sign: $0.771 + 0.636i$
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: \((2,\ 3234,\ (\ :1/2),\ 0.771 + 0.636i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3885278307\)
\(L(\frac12)\) \(\approx\) \(0.3885278307\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.48 + 2.19i)T \)
good5 \( 1 - 2.84iT - 5T^{2} \)
13 \( 1 - 0.726T + 13T^{2} \)
17 \( 1 + 5.60T + 17T^{2} \)
19 \( 1 + 6.09T + 19T^{2} \)
23 \( 1 - 2.98T + 23T^{2} \)
29 \( 1 + 3.95iT - 29T^{2} \)
31 \( 1 + 10.0iT - 31T^{2} \)
37 \( 1 + 2.70T + 37T^{2} \)
41 \( 1 + 6.88T + 41T^{2} \)
43 \( 1 + 5.82iT - 43T^{2} \)
47 \( 1 - 6.81iT - 47T^{2} \)
53 \( 1 - 3.78T + 53T^{2} \)
59 \( 1 + 12.4iT - 59T^{2} \)
61 \( 1 + 1.34T + 61T^{2} \)
67 \( 1 + 14.7T + 67T^{2} \)
71 \( 1 + 1.05T + 71T^{2} \)
73 \( 1 + 13.8T + 73T^{2} \)
79 \( 1 + 8.51iT - 79T^{2} \)
83 \( 1 + 6.34T + 83T^{2} \)
89 \( 1 + 4.47iT - 89T^{2} \)
97 \( 1 - 4.27iT - 97T^{2} \)
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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

−8.705849541276141064596207894468, −7.76884463518532479047465349536, −6.87727882057530750152074777170, −6.35694443088457015039330625847, −5.84938324862691619108068395829, −4.58982209136959979114419831249, −4.00039113681630247717844904734, −3.13413381545582066961436658628, −2.11648130361968479559109077687, −0.11940734200354774937074580390, 1.28656137549361924704856962025, 1.83660333431070211987231689287, 3.02064439435748523155022110827, 4.24798273114793556384624469867, 4.65616498485999462123958665018, 5.54398429098858805556647632064, 6.63842790603606103443357201807, 7.11922894453544292036900761348, 8.489747711549021778067260898473, 8.701681771350290538349013706633

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