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Properties

Label 2-1856-29.3-c0-0-0
Degree $2$
Conductor $1856$
Sign $0.947 - 0.318i$
Analytic cond. $0.926264$
Root an. cond. $0.962426$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.52 − 0.347i)5-s + (−0.781 + 0.623i)9-s + (1.40 + 1.12i)13-s + (−1.19 + 1.19i)17-s + (1.30 − 0.626i)25-s + (0.781 + 0.623i)29-s + (0.211 − 1.87i)37-s + (−0.467 − 0.467i)41-s + (−0.974 + 1.22i)45-s + (−0.623 − 0.781i)49-s + (−0.433 − 1.90i)53-s + (−1.00 + 0.351i)61-s + (2.53 + 1.22i)65-s + (0.900 − 0.566i)73-s + (0.222 − 0.974i)81-s + ⋯
L(s)  = 1  + (1.52 − 0.347i)5-s + (−0.781 + 0.623i)9-s + (1.40 + 1.12i)13-s + (−1.19 + 1.19i)17-s + (1.30 − 0.626i)25-s + (0.781 + 0.623i)29-s + (0.211 − 1.87i)37-s + (−0.467 − 0.467i)41-s + (−0.974 + 1.22i)45-s + (−0.623 − 0.781i)49-s + (−0.433 − 1.90i)53-s + (−1.00 + 0.351i)61-s + (2.53 + 1.22i)65-s + (0.900 − 0.566i)73-s + (0.222 − 0.974i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1856\)    =    \(2^{6} \cdot 29\)
Sign: $0.947 - 0.318i$
Analytic conductor: \(0.926264\)
Root analytic conductor: \(0.962426\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1856} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1856,\ (\ :0),\ 0.947 - 0.318i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.446735768\)
\(L(\frac12)\) \(\approx\) \(1.446735768\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
29 \( 1 + (-0.781 - 0.623i)T \)
good3 \( 1 + (0.781 - 0.623i)T^{2} \)
5 \( 1 + (-1.52 + 0.347i)T + (0.900 - 0.433i)T^{2} \)
7 \( 1 + (0.623 + 0.781i)T^{2} \)
11 \( 1 + (-0.974 + 0.222i)T^{2} \)
13 \( 1 + (-1.40 - 1.12i)T + (0.222 + 0.974i)T^{2} \)
17 \( 1 + (1.19 - 1.19i)T - iT^{2} \)
19 \( 1 + (-0.781 - 0.623i)T^{2} \)
23 \( 1 + (-0.900 - 0.433i)T^{2} \)
31 \( 1 + (0.433 + 0.900i)T^{2} \)
37 \( 1 + (-0.211 + 1.87i)T + (-0.974 - 0.222i)T^{2} \)
41 \( 1 + (0.467 + 0.467i)T + iT^{2} \)
43 \( 1 + (-0.433 + 0.900i)T^{2} \)
47 \( 1 + (0.974 - 0.222i)T^{2} \)
53 \( 1 + (0.433 + 1.90i)T + (-0.900 + 0.433i)T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + (1.00 - 0.351i)T + (0.781 - 0.623i)T^{2} \)
67 \( 1 + (0.222 - 0.974i)T^{2} \)
71 \( 1 + (0.222 + 0.974i)T^{2} \)
73 \( 1 + (-0.900 + 0.566i)T + (0.433 - 0.900i)T^{2} \)
79 \( 1 + (0.974 + 0.222i)T^{2} \)
83 \( 1 + (0.623 - 0.781i)T^{2} \)
89 \( 1 + (-0.189 - 0.119i)T + (0.433 + 0.900i)T^{2} \)
97 \( 1 + (1.78 + 0.623i)T + (0.781 + 0.623i)T^{2} \)
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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

−9.259157278812029339928805610004, −8.791991420073749142629675678276, −8.237008571120117411962670147365, −6.79173850058173820433147019438, −6.22940830894889911421718382577, −5.58717290993957827123351594990, −4.68936254415127510125396337225, −3.63111481254440786269220672511, −2.22155285137095211693284232990, −1.67022329610168924545175541698, 1.22759390796532330911348648127, 2.64059303162516570119059444128, 3.15142000164321402427634108053, 4.61225561192557979988744794547, 5.57199592016837486618885346039, 6.26415210185701133981019729714, 6.62317136451451075042102336173, 7.976970957388671984012416818210, 8.745786701966357768446015466789, 9.398595932960067641976639128864

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