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Properties

Label 2-1856-29.14-c0-0-0
Degree $2$
Conductor $1856$
Sign $0.0384 + 0.999i$
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  + (−0.678 − 0.541i)5-s + (0.433 − 0.900i)9-s + (−0.193 − 0.400i)13-s + (−0.158 + 0.158i)17-s + (−0.0549 − 0.240i)25-s + (−0.433 − 0.900i)29-s + (−0.351 − 1.00i)37-s + (−1.19 − 1.19i)41-s + (−0.781 + 0.376i)45-s + (0.900 + 0.433i)49-s + (0.974 − 1.22i)53-s + (1.05 + 1.68i)61-s + (−0.0859 + 0.376i)65-s + (0.222 − 1.97i)73-s + (−0.623 − 0.781i)81-s + ⋯
L(s)  = 1  + (−0.678 − 0.541i)5-s + (0.433 − 0.900i)9-s + (−0.193 − 0.400i)13-s + (−0.158 + 0.158i)17-s + (−0.0549 − 0.240i)25-s + (−0.433 − 0.900i)29-s + (−0.351 − 1.00i)37-s + (−1.19 − 1.19i)41-s + (−0.781 + 0.376i)45-s + (0.900 + 0.433i)49-s + (0.974 − 1.22i)53-s + (1.05 + 1.68i)61-s + (−0.0859 + 0.376i)65-s + (0.222 − 1.97i)73-s + (−0.623 − 0.781i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0384 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0384 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1856\)    =    \(2^{6} \cdot 29\)
Sign: $0.0384 + 0.999i$
Analytic conductor: \(0.926264\)
Root analytic conductor: \(0.962426\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1856} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1856,\ (\ :0),\ 0.0384 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8936880692\)
\(L(\frac12)\) \(\approx\) \(0.8936880692\)
\(L(1)\) 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 \( 1 + (0.433 + 0.900i)T \)
good3 \( 1 + (-0.433 + 0.900i)T^{2} \)
5 \( 1 + (0.678 + 0.541i)T + (0.222 + 0.974i)T^{2} \)
7 \( 1 + (-0.900 - 0.433i)T^{2} \)
11 \( 1 + (-0.781 - 0.623i)T^{2} \)
13 \( 1 + (0.193 + 0.400i)T + (-0.623 + 0.781i)T^{2} \)
17 \( 1 + (0.158 - 0.158i)T - iT^{2} \)
19 \( 1 + (0.433 + 0.900i)T^{2} \)
23 \( 1 + (-0.222 + 0.974i)T^{2} \)
31 \( 1 + (-0.974 + 0.222i)T^{2} \)
37 \( 1 + (0.351 + 1.00i)T + (-0.781 + 0.623i)T^{2} \)
41 \( 1 + (1.19 + 1.19i)T + iT^{2} \)
43 \( 1 + (0.974 + 0.222i)T^{2} \)
47 \( 1 + (0.781 + 0.623i)T^{2} \)
53 \( 1 + (-0.974 + 1.22i)T + (-0.222 - 0.974i)T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + (-1.05 - 1.68i)T + (-0.433 + 0.900i)T^{2} \)
67 \( 1 + (-0.623 - 0.781i)T^{2} \)
71 \( 1 + (-0.623 + 0.781i)T^{2} \)
73 \( 1 + (-0.222 + 1.97i)T + (-0.974 - 0.222i)T^{2} \)
79 \( 1 + (0.781 - 0.623i)T^{2} \)
83 \( 1 + (-0.900 + 0.433i)T^{2} \)
89 \( 1 + (-0.0739 - 0.656i)T + (-0.974 + 0.222i)T^{2} \)
97 \( 1 + (0.566 - 0.900i)T + (-0.433 - 0.900i)T^{2} \)
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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

−9.102584232611001222873239383713, −8.545272460723507888859521694305, −7.65758771837954011731189858218, −6.98332227155333298552529599245, −6.03497605736800262927128915752, −5.14804235182431153331727995335, −4.12188760537594805231477497240, −3.59685801536559399976602073269, −2.19942098767249335965380954622, −0.67935529348975995626076149235, 1.66850376985658439708197506329, 2.83675853314938403282751512574, 3.80294095560299264897744910979, 4.70419616125204569626702922123, 5.48907405354531979094523546939, 6.75667811519820043192690689359, 7.18499768996871301724622297040, 8.007664384933425504494952666578, 8.713682457074323771039784888433, 9.739657159112394784368311855735

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