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Properties

Label 2-1856-29.15-c0-0-0
Degree $2$
Conductor $1856$
Sign $0.856 - 0.516i$
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 + (1.40 + 1.40i)17-s + (−0.0549 − 0.240i)25-s + (0.433 + 0.900i)29-s + (1.59 − 0.559i)37-s + (0.752 − 0.752i)41-s + (0.781 − 0.376i)45-s + (0.900 + 0.433i)49-s + (−0.974 + 1.22i)53-s + (0.189 − 0.119i)61-s + (0.0859 − 0.376i)65-s + (0.222 + 0.0250i)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 + (1.40 + 1.40i)17-s + (−0.0549 − 0.240i)25-s + (0.433 + 0.900i)29-s + (1.59 − 0.559i)37-s + (0.752 − 0.752i)41-s + (0.781 − 0.376i)45-s + (0.900 + 0.433i)49-s + (−0.974 + 1.22i)53-s + (0.189 − 0.119i)61-s + (0.0859 − 0.376i)65-s + (0.222 + 0.0250i)73-s + (−0.623 − 0.781i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.856 - 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.856 - 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1856\)    =    \(2^{6} \cdot 29\)
Sign: $0.856 - 0.516i$
Analytic conductor: \(0.926264\)
Root analytic conductor: \(0.962426\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1856} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1856,\ (\ :0),\ 0.856 - 0.516i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9918325694\)
\(L(\frac12)\) \(\approx\) \(0.9918325694\)
\(L(1)\) 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 \( 1 + (-1.40 - 1.40i)T + iT^{2} \)
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 \( 1 + (-1.59 + 0.559i)T + (0.781 - 0.623i)T^{2} \)
41 \( 1 + (-0.752 + 0.752i)T - iT^{2} \)
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 \( 1 + (-0.189 + 0.119i)T + (0.433 - 0.900i)T^{2} \)
67 \( 1 + (-0.623 - 0.781i)T^{2} \)
71 \( 1 + (-0.623 + 0.781i)T^{2} \)
73 \( 1 + (-0.222 - 0.0250i)T + (0.974 + 0.222i)T^{2} \)
79 \( 1 + (-0.781 + 0.623i)T^{2} \)
83 \( 1 + (-0.900 + 0.433i)T^{2} \)
89 \( 1 + (1.87 - 0.211i)T + (0.974 - 0.222i)T^{2} \)
97 \( 1 + (1.43 + 0.900i)T + (0.433 + 0.900i)T^{2} \)
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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

−9.393132234368516040303280478530, −8.548675152081692139489661343921, −7.990291542796853528151912916170, −7.40577331014251191736344295989, −6.15883566027178849582303380430, −5.50201259027812312081515762302, −4.50204285419557493765869477516, −3.79722078432629607299618389745, −2.62693037745701761279060378870, −1.30609776051735424181152761880, 0.868046967473695138475411780637, 2.77050554572069811797245974033, 3.32858372658675056549833515028, 4.30607128305848743696133299953, 5.42453035359269117140177072803, 6.18739653862587337432232827286, 7.09059352510724518759700615924, 7.77268781838790362610809992695, 8.446440603506879860410981755960, 9.606992605294654343686926128016

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