Properties

Label 2-1856-29.28-c1-0-35
Degree $2$
Conductor $1856$
Sign $0.557 + 0.830i$
Analytic cond. $14.8202$
Root an. cond. $3.84970$
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·3-s − 5-s + 4.47·7-s + 2·9-s − 3i·11-s + 13-s + i·15-s − 4.47i·17-s + 4i·19-s − 4.47i·21-s + 4.47·23-s − 4·25-s − 5i·27-s + (3 + 4.47i)29-s + 5i·31-s + ⋯
L(s)  = 1  − 0.577i·3-s − 0.447·5-s + 1.69·7-s + 0.666·9-s − 0.904i·11-s + 0.277·13-s + 0.258i·15-s − 1.08i·17-s + 0.917i·19-s − 0.975i·21-s + 0.932·23-s − 0.800·25-s − 0.962i·27-s + (0.557 + 0.830i)29-s + 0.898i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.557 + 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.557 + 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1856\)    =    \(2^{6} \cdot 29\)
Sign: $0.557 + 0.830i$
Analytic conductor: \(14.8202\)
Root analytic conductor: \(3.84970\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1856} (1217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1856,\ (\ :1/2),\ 0.557 + 0.830i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.156720112\)
\(L(\frac12)\) \(\approx\) \(2.156720112\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
29 \( 1 + (-3 - 4.47i)T \)
good3 \( 1 + iT - 3T^{2} \)
5 \( 1 + T + 5T^{2} \)
7 \( 1 - 4.47T + 7T^{2} \)
11 \( 1 + 3iT - 11T^{2} \)
13 \( 1 - T + 13T^{2} \)
17 \( 1 + 4.47iT - 17T^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 - 4.47T + 23T^{2} \)
31 \( 1 - 5iT - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 4.47iT - 41T^{2} \)
43 \( 1 - iT - 43T^{2} \)
47 \( 1 - 3iT - 47T^{2} \)
53 \( 1 - 9T + 53T^{2} \)
59 \( 1 + 4.47T + 59T^{2} \)
61 \( 1 + 13.4iT - 61T^{2} \)
67 \( 1 + 8.94T + 67T^{2} \)
71 \( 1 + 8.94T + 71T^{2} \)
73 \( 1 - 8.94iT - 73T^{2} \)
79 \( 1 - 15iT - 79T^{2} \)
83 \( 1 - 4.47T + 83T^{2} \)
89 \( 1 + 13.4iT - 89T^{2} \)
97 \( 1 + 13.4iT - 97T^{2} \)
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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

−8.778037233739535162727708615895, −8.310243529383239306640141222226, −7.51858842580402130956155962917, −7.03443261374733835515707021697, −5.85934883382914454728278371033, −5.01548351931013075866145275791, −4.25410893852904764719896235974, −3.15058533608362532325851652377, −1.80645496015045977964939515932, −0.971158506903097112317827998623, 1.28999199747980159103584081060, 2.29563519122987647899978825162, 3.83964071288135945493792110630, 4.48599381485647143423684282332, 4.96119961592316698842263259113, 6.10442342874628735167884510683, 7.29413593535235948786089244216, 7.72533688668088961363755364575, 8.588579718055368047586563830403, 9.298048171558416843681043185997

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