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Properties

Label 2-3234-77.76-c1-0-42
Degree $2$
Conductor $3234$
Sign $0.798 - 0.601i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.798 - 0.601i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.798 - 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3234\)    =    \(2 \cdot 3 \cdot 7^{2} \cdot 11\)
Sign: $0.798 - 0.601i$
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: \((2,\ 3234,\ (\ :1/2),\ 0.798 - 0.601i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.403346135\)
\(L(\frac12)\) \(\approx\) \(1.403346135\)
\(L(\frac{3}{2})\) 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

−8.810850222267719940634566526905, −8.028063249969558619883016427553, −7.06354822116300090326175542575, −6.61301003150114946898053568684, −5.64381445692624576478747390096, −5.02264471489442295490871026212, −4.15410123242802421648431028362, −3.31370918487764910748590081907, −2.35575920017437445960021986829, −0.55487955351336026486162839218, 0.904018424139354618623068360577, 1.96719987783428621593002963155, 2.69599020465437116568757434193, 3.81031121008218434071995145292, 4.82251862077500897525384777232, 5.19319107637051715079199146098, 6.52041660288126132460272628696, 7.08829515674424552196520914341, 7.82861868975490771177174533425, 8.827638007182748632844297519085

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