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Properties

Label 2-1856-29.19-c0-0-0
Degree $2$
Conductor $1856$
Sign $0.857 - 0.514i$
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  + (1.52 + 0.347i)5-s + (0.781 + 0.623i)9-s + (−1.40 + 1.12i)13-s + (0.752 − 0.752i)17-s + (1.30 + 0.626i)25-s + (−0.781 + 0.623i)29-s + (−0.656 + 0.0739i)37-s + (−1.33 − 1.33i)41-s + (0.974 + 1.22i)45-s + (−0.623 + 0.781i)49-s + (0.433 − 1.90i)53-s + (0.559 − 1.59i)61-s + (−2.53 + 1.22i)65-s + (0.900 − 1.43i)73-s + (0.222 + 0.974i)81-s + ⋯
L(s)  = 1  + (1.52 + 0.347i)5-s + (0.781 + 0.623i)9-s + (−1.40 + 1.12i)13-s + (0.752 − 0.752i)17-s + (1.30 + 0.626i)25-s + (−0.781 + 0.623i)29-s + (−0.656 + 0.0739i)37-s + (−1.33 − 1.33i)41-s + (0.974 + 1.22i)45-s + (−0.623 + 0.781i)49-s + (0.433 − 1.90i)53-s + (0.559 − 1.59i)61-s + (−2.53 + 1.22i)65-s + (0.900 − 1.43i)73-s + (0.222 + 0.974i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 - 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 - 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1856\)    =    \(2^{6} \cdot 29\)
Sign: $0.857 - 0.514i$
Analytic conductor: \(0.926264\)
Root analytic conductor: \(0.962426\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1856} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1856,\ (\ :0),\ 0.857 - 0.514i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.496542999\)
\(L(\frac12)\) \(\approx\) \(1.496542999\)
\(L(1)\) 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 \( 1 + (0.781 - 0.623i)T \)
good3 \( 1 + (-0.781 - 0.623i)T^{2} \)
5 \( 1 + (-1.52 - 0.347i)T + (0.900 + 0.433i)T^{2} \)
7 \( 1 + (0.623 - 0.781i)T^{2} \)
11 \( 1 + (0.974 + 0.222i)T^{2} \)
13 \( 1 + (1.40 - 1.12i)T + (0.222 - 0.974i)T^{2} \)
17 \( 1 + (-0.752 + 0.752i)T - iT^{2} \)
19 \( 1 + (0.781 - 0.623i)T^{2} \)
23 \( 1 + (-0.900 + 0.433i)T^{2} \)
31 \( 1 + (-0.433 + 0.900i)T^{2} \)
37 \( 1 + (0.656 - 0.0739i)T + (0.974 - 0.222i)T^{2} \)
41 \( 1 + (1.33 + 1.33i)T + iT^{2} \)
43 \( 1 + (0.433 + 0.900i)T^{2} \)
47 \( 1 + (-0.974 - 0.222i)T^{2} \)
53 \( 1 + (-0.433 + 1.90i)T + (-0.900 - 0.433i)T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + (-0.559 + 1.59i)T + (-0.781 - 0.623i)T^{2} \)
67 \( 1 + (0.222 + 0.974i)T^{2} \)
71 \( 1 + (0.222 - 0.974i)T^{2} \)
73 \( 1 + (-0.900 + 1.43i)T + (-0.433 - 0.900i)T^{2} \)
79 \( 1 + (-0.974 + 0.222i)T^{2} \)
83 \( 1 + (0.623 + 0.781i)T^{2} \)
89 \( 1 + (-1.05 - 1.68i)T + (-0.433 + 0.900i)T^{2} \)
97 \( 1 + (0.218 + 0.623i)T + (-0.781 + 0.623i)T^{2} \)
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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

−9.632560319087923702526564682809, −9.019397399257344298115345763767, −7.74931283995296260648297458852, −7.00965529726659132643101886825, −6.49237372493685082391101846581, −5.12700631925494245668625718555, −5.07552135731816097118394820481, −3.56331741495776058511837354229, −2.27448776592897472172498474750, −1.77824934573855708675068082446, 1.27135742673913771491527722652, 2.29088214902001502614399833689, 3.37801666994883554862807514778, 4.61661126904549965015475251594, 5.44923525998251875558629041007, 6.01279891676909728253590419640, 6.96062433769143550892691903362, 7.75555853963149264991444801350, 8.715661226837158061175401252598, 9.602772037862711844058266652093

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