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Properties

Label 2-1856-116.7-c0-0-0
Degree $2$
Conductor $1856$
Sign $0.626 - 0.779i$
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  + (0.277 + 1.21i)5-s + (0.623 + 0.781i)9-s + (1.12 − 1.40i)13-s − 0.445·17-s + (−0.499 + 0.240i)25-s + (−0.623 + 0.781i)29-s + (1.12 + 1.40i)37-s − 1.80·41-s + (−0.777 + 0.974i)45-s + (0.623 + 0.781i)49-s + (−0.0990 − 0.433i)53-s + (−0.400 − 0.193i)61-s + (2.02 + 0.974i)65-s + (0.0990 − 0.433i)73-s + (−0.222 + 0.974i)81-s + ⋯
L(s)  = 1  + (0.277 + 1.21i)5-s + (0.623 + 0.781i)9-s + (1.12 − 1.40i)13-s − 0.445·17-s + (−0.499 + 0.240i)25-s + (−0.623 + 0.781i)29-s + (1.12 + 1.40i)37-s − 1.80·41-s + (−0.777 + 0.974i)45-s + (0.623 + 0.781i)49-s + (−0.0990 − 0.433i)53-s + (−0.400 − 0.193i)61-s + (2.02 + 0.974i)65-s + (0.0990 − 0.433i)73-s + (−0.222 + 0.974i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.626 - 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.626 - 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1856\)    =    \(2^{6} \cdot 29\)
Sign: $0.626 - 0.779i$
Analytic conductor: \(0.926264\)
Root analytic conductor: \(0.962426\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1856} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1856,\ (\ :0),\ 0.626 - 0.779i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.288809278\)
\(L(\frac12)\) \(\approx\) \(1.288809278\)
\(L(1)\) 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 \( 1 + (0.623 - 0.781i)T \)
good3 \( 1 + (-0.623 - 0.781i)T^{2} \)
5 \( 1 + (-0.277 - 1.21i)T + (-0.900 + 0.433i)T^{2} \)
7 \( 1 + (-0.623 - 0.781i)T^{2} \)
11 \( 1 + (0.222 + 0.974i)T^{2} \)
13 \( 1 + (-1.12 + 1.40i)T + (-0.222 - 0.974i)T^{2} \)
17 \( 1 + 0.445T + T^{2} \)
19 \( 1 + (-0.623 + 0.781i)T^{2} \)
23 \( 1 + (0.900 + 0.433i)T^{2} \)
31 \( 1 + (0.900 - 0.433i)T^{2} \)
37 \( 1 + (-1.12 - 1.40i)T + (-0.222 + 0.974i)T^{2} \)
41 \( 1 + 1.80T + T^{2} \)
43 \( 1 + (0.900 + 0.433i)T^{2} \)
47 \( 1 + (0.222 + 0.974i)T^{2} \)
53 \( 1 + (0.0990 + 0.433i)T + (-0.900 + 0.433i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (0.400 + 0.193i)T + (0.623 + 0.781i)T^{2} \)
67 \( 1 + (0.222 - 0.974i)T^{2} \)
71 \( 1 + (0.222 + 0.974i)T^{2} \)
73 \( 1 + (-0.0990 + 0.433i)T + (-0.900 - 0.433i)T^{2} \)
79 \( 1 + (0.222 - 0.974i)T^{2} \)
83 \( 1 + (-0.623 + 0.781i)T^{2} \)
89 \( 1 + (0.277 + 1.21i)T + (-0.900 + 0.433i)T^{2} \)
97 \( 1 + (-1.62 + 0.781i)T + (0.623 - 0.781i)T^{2} \)
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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

−9.781983020766764692772453938604, −8.625457819933930868442899742051, −7.933718158658665407792342151383, −7.14740192313709958699333247571, −6.41761946360911394575421307491, −5.63879461079634537554409704164, −4.66876623497616069309601969618, −3.48318108715966042457139637356, −2.79443863694213147961325159324, −1.57944896738247071962789934701, 1.11556240797236851068877923294, 2.06646359604052975073255466790, 3.77785340471309766806527586473, 4.25592842982005032787899473162, 5.23672725080854491044571008894, 6.19154268177768072876828118099, 6.80350393593685411112774899811, 7.85039414139200600374613869944, 8.898157756077322637884659678138, 9.076112476034306551072650011050

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