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Properties

Label 2-2800-112.67-c0-0-1
Degree $2$
Conductor $2800$
Sign $0.981 + 0.193i$
Analytic cond. $1.39738$
Root an. cond. $1.18210$
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  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.5 + 0.866i)7-s + 0.999·8-s + (0.866 − 0.5i)9-s + (1.36 − 0.366i)11-s + (1 + i)13-s + (0.499 − 0.866i)14-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + (−0.866 − 0.499i)18-s + (−1.36 − 0.366i)19-s + (−1 − 0.999i)22-s + (0.5 + 0.866i)23-s + (0.366 − 1.36i)26-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.5 + 0.866i)7-s + 0.999·8-s + (0.866 − 0.5i)9-s + (1.36 − 0.366i)11-s + (1 + i)13-s + (0.499 − 0.866i)14-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + (−0.866 − 0.499i)18-s + (−1.36 − 0.366i)19-s + (−1 − 0.999i)22-s + (0.5 + 0.866i)23-s + (0.366 − 1.36i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2800\)    =    \(2^{4} \cdot 5^{2} \cdot 7\)
Sign: $0.981 + 0.193i$
Analytic conductor: \(1.39738\)
Root analytic conductor: \(1.18210\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2800} (851, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2800,\ (\ :0),\ 0.981 + 0.193i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.117449185\)
\(L(\frac12)\) \(\approx\) \(1.117449185\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 \)
7 \( 1 + (-0.5 - 0.866i)T \)
good3 \( 1 + (-0.866 + 0.5i)T^{2} \)
11 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
13 \( 1 + (-1 - i)T + iT^{2} \)
17 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + iT^{2} \)
31 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.866 + 0.5i)T^{2} \)
41 \( 1 - iT - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.866 + 0.5i)T^{2} \)
59 \( 1 + (0.866 - 0.5i)T^{2} \)
61 \( 1 + (-0.866 - 0.5i)T^{2} \)
67 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
71 \( 1 - T + T^{2} \)
73 \( 1 + (1.73 + i)T + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-1 + i)T - iT^{2} \)
89 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 - T + T^{2} \)
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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

−9.043749685508232663348699316574, −8.606280904113281810761363030145, −7.66499832346478871292062684027, −6.57734226654672914737289477851, −6.16928920910490700307105674088, −4.67331732396956550210901720136, −4.07568978150906967576894867280, −3.36560619816282058173942571595, −1.89409176778849166662432496750, −1.46542866673449435389175457304, 1.00683194900651601988550359906, 1.93212231722695185719114895404, 3.78762177262275605826817258762, 4.38579513124886418957397369064, 5.13467009167836695809356820775, 6.21203888273467235323890751369, 6.92192930697907253509914720226, 7.33213860634457476435270648277, 8.328009783891268211173559627483, 8.755074926305222118705545495637

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