Properties

Label 2-1856-1.1-c3-0-79
Degree $2$
Conductor $1856$
Sign $1$
Analytic cond. $109.507$
Root an. cond. $10.4645$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8.07·3-s − 14.4·5-s + 28.4·7-s + 38.2·9-s − 36.6·11-s + 87.3·13-s − 116.·15-s − 97.2·17-s + 95.0·19-s + 229.·21-s + 120.·23-s + 82.6·25-s + 91.0·27-s + 29·29-s − 124.·31-s − 295.·33-s − 410.·35-s + 164.·37-s + 705.·39-s − 423.·41-s + 450.·43-s − 551.·45-s + 487.·47-s + 467.·49-s − 785.·51-s − 655.·53-s + 527.·55-s + ⋯
L(s)  = 1  + 1.55·3-s − 1.28·5-s + 1.53·7-s + 1.41·9-s − 1.00·11-s + 1.86·13-s − 2.00·15-s − 1.38·17-s + 1.14·19-s + 2.38·21-s + 1.09·23-s + 0.661·25-s + 0.648·27-s + 0.185·29-s − 0.718·31-s − 1.56·33-s − 1.98·35-s + 0.730·37-s + 2.89·39-s − 1.61·41-s + 1.59·43-s − 1.82·45-s + 1.51·47-s + 1.36·49-s − 2.15·51-s − 1.69·53-s + 1.29·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1856\)    =    \(2^{6} \cdot 29\)
Sign: $1$
Analytic conductor: \(109.507\)
Root analytic conductor: \(10.4645\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1856,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(4.097106200\)
\(L(\frac12)\) \(\approx\) \(4.097106200\)
\(L(\frac{5}{2})\) 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 \( 1 - 29T \)
good3 \( 1 - 8.07T + 27T^{2} \)
5 \( 1 + 14.4T + 125T^{2} \)
7 \( 1 - 28.4T + 343T^{2} \)
11 \( 1 + 36.6T + 1.33e3T^{2} \)
13 \( 1 - 87.3T + 2.19e3T^{2} \)
17 \( 1 + 97.2T + 4.91e3T^{2} \)
19 \( 1 - 95.0T + 6.85e3T^{2} \)
23 \( 1 - 120.T + 1.21e4T^{2} \)
31 \( 1 + 124.T + 2.97e4T^{2} \)
37 \( 1 - 164.T + 5.06e4T^{2} \)
41 \( 1 + 423.T + 6.89e4T^{2} \)
43 \( 1 - 450.T + 7.95e4T^{2} \)
47 \( 1 - 487.T + 1.03e5T^{2} \)
53 \( 1 + 655.T + 1.48e5T^{2} \)
59 \( 1 - 436.T + 2.05e5T^{2} \)
61 \( 1 - 432.T + 2.26e5T^{2} \)
67 \( 1 - 220.T + 3.00e5T^{2} \)
71 \( 1 - 152.T + 3.57e5T^{2} \)
73 \( 1 + 527.T + 3.89e5T^{2} \)
79 \( 1 - 445.T + 4.93e5T^{2} \)
83 \( 1 - 452.T + 5.71e5T^{2} \)
89 \( 1 - 779.T + 7.04e5T^{2} \)
97 \( 1 + 544.T + 9.12e5T^{2} \)
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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

−8.708808125698409048846100103775, −8.060167339719847030461785299307, −7.74448487216580796167537796613, −6.93069156492268320158816037785, −5.43887506213100488258167326379, −4.48865247565282382290821773711, −3.82115182109808869526879187793, −3.01866766945827207018038147375, −1.99316746129999197662103137795, −0.917736445556294524802415591992, 0.917736445556294524802415591992, 1.99316746129999197662103137795, 3.01866766945827207018038147375, 3.82115182109808869526879187793, 4.48865247565282382290821773711, 5.43887506213100488258167326379, 6.93069156492268320158816037785, 7.74448487216580796167537796613, 8.060167339719847030461785299307, 8.708808125698409048846100103775

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