Properties

Label 2-3234-77.76-c1-0-47
Degree $2$
Conductor $3234$
Sign $0.813 + 0.581i$
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 − 0.942i·5-s + 6-s i·8-s − 9-s + 0.942·10-s + (1.67 + 2.86i)11-s + i·12-s − 2.81·13-s − 0.942·15-s + 16-s − 4.30·17-s i·18-s − 2.51·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.421i·5-s + 0.408·6-s − 0.353i·8-s − 0.333·9-s + 0.298·10-s + (0.503 + 0.863i)11-s + 0.288i·12-s − 0.781·13-s − 0.243·15-s + 0.250·16-s − 1.04·17-s − 0.235i·18-s − 0.577·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 + 0.581i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.813 + 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3234\)    =    \(2 \cdot 3 \cdot 7^{2} \cdot 11\)
Sign: $0.813 + 0.581i$
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: \((2,\ 3234,\ (\ :1/2),\ 0.813 + 0.581i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.406672550\)
\(L(\frac12)\) \(\approx\) \(1.406672550\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.67 - 2.86i)T \)
good5 \( 1 + 0.942iT - 5T^{2} \)
13 \( 1 + 2.81T + 13T^{2} \)
17 \( 1 + 4.30T + 17T^{2} \)
19 \( 1 + 2.51T + 19T^{2} \)
23 \( 1 - 4.33T + 23T^{2} \)
29 \( 1 + 8.01iT - 29T^{2} \)
31 \( 1 - 6.05iT - 31T^{2} \)
37 \( 1 - 10.4T + 37T^{2} \)
41 \( 1 - 5.20T + 41T^{2} \)
43 \( 1 + 11.3iT - 43T^{2} \)
47 \( 1 - 8.96iT - 47T^{2} \)
53 \( 1 - 4.55T + 53T^{2} \)
59 \( 1 + 13.4iT - 59T^{2} \)
61 \( 1 - 2.41T + 61T^{2} \)
67 \( 1 + 2.22T + 67T^{2} \)
71 \( 1 + 4.80T + 71T^{2} \)
73 \( 1 + 4.63T + 73T^{2} \)
79 \( 1 + 9.26iT - 79T^{2} \)
83 \( 1 + 8.84T + 83T^{2} \)
89 \( 1 + 11.5iT - 89T^{2} \)
97 \( 1 + 12.5iT - 97T^{2} \)
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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

−8.565562184862029480810800275819, −7.68154980094471585442934106513, −7.05977228645128359892677739376, −6.51439502783390417524441778866, −5.67314821459338556848835173544, −4.64372004116583813659836042185, −4.32568415974254729071700936041, −2.83566276367500264101341994399, −1.85884044551487565249091622406, −0.53580444276612863715454343667, 0.958361605904221417194793974617, 2.45681020922217355147624025071, 3.00149042312010352916054418484, 4.05061027234300528682076389145, 4.63241767504296852696178947607, 5.57050530579704394650476499332, 6.43980003250442222864985223913, 7.20631460936215578076524907283, 8.246243068693301028844878780884, 8.981722137723648800217736078486

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