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Properties

Label 2-1879-1879.1878-c0-0-1
Degree $2$
Conductor $1879$
Sign $1$
Analytic cond. $0.937743$
Root an. cond. $0.968371$
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  − 1.41i·3-s − 4-s − 5-s + 7-s − 1.00·9-s + 1.41i·11-s + 1.41i·12-s + 1.41i·13-s + 1.41i·15-s + 16-s + 17-s + 1.41i·19-s + 20-s − 1.41i·21-s − 28-s − 29-s + ⋯
L(s)  = 1  − 1.41i·3-s − 4-s − 5-s + 7-s − 1.00·9-s + 1.41i·11-s + 1.41i·12-s + 1.41i·13-s + 1.41i·15-s + 16-s + 17-s + 1.41i·19-s + 20-s − 1.41i·21-s − 28-s − 29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1879 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1879 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1879\)
Sign: $1$
Analytic conductor: \(0.937743\)
Root analytic conductor: \(0.968371\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1879} (1878, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1879,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7749976186\)
\(L(\frac12)\) \(\approx\) \(0.7749976186\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1879 \( 1 - T \)
good2 \( 1 + T^{2} \)
3 \( 1 + 1.41iT - T^{2} \)
5 \( 1 + T + T^{2} \)
7 \( 1 - T + T^{2} \)
11 \( 1 - 1.41iT - T^{2} \)
13 \( 1 - 1.41iT - T^{2} \)
17 \( 1 - T + T^{2} \)
19 \( 1 - 1.41iT - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + T + T^{2} \)
31 \( 1 + 1.41iT - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - 1.41iT - T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 - T + T^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - T + T^{2} \)
71 \( 1 - T + T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 - 1.41iT - T^{2} \)
83 \( 1 + T + T^{2} \)
89 \( 1 + T + T^{2} \)
97 \( 1 - 2T + T^{2} \)
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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

−9.370399634060871375902868569142, −8.234280593587662279122451104833, −7.74779662760334829688882171809, −7.48435870685148663159202296178, −6.37328640108264964546843844888, −5.35970841940934843052341151416, −4.33578976429322474977873793521, −3.88201678605978601365126854261, −2.09716192503186392959750944838, −1.29991281310433338299942838383, 0.69757154387950460782827717633, 3.17910707404820514060068409827, 3.61746357059949234659348580080, 4.48903361712022697442449558081, 5.25629185899777265106673310416, 5.63733187887800325295064240596, 7.43985101381334695271036746798, 8.109264745607319837110190017322, 8.683770632871360884355417476284, 9.283963555696668627479307806680

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