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Properties

Label 2-2200-440.307-c0-0-3
Degree $2$
Conductor $2200$
Sign $0.973 - 0.229i$
Analytic cond. $1.09794$
Root an. cond. $1.04782$
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  + (−0.707 − 0.707i)2-s + 1.00i·4-s + (1.41 + 1.41i)7-s + (0.707 − 0.707i)8-s + i·9-s − 11-s + (1.41 − 1.41i)13-s − 2.00i·14-s − 1.00·16-s + (0.707 − 0.707i)18-s + (0.707 + 0.707i)22-s − 2.00·26-s + (−1.41 + 1.41i)28-s + (0.707 + 0.707i)32-s − 1.00·36-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + 1.00i·4-s + (1.41 + 1.41i)7-s + (0.707 − 0.707i)8-s + i·9-s − 11-s + (1.41 − 1.41i)13-s − 2.00i·14-s − 1.00·16-s + (0.707 − 0.707i)18-s + (0.707 + 0.707i)22-s − 2.00·26-s + (−1.41 + 1.41i)28-s + (0.707 + 0.707i)32-s − 1.00·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2200\)    =    \(2^{3} \cdot 5^{2} \cdot 11\)
Sign: $0.973 - 0.229i$
Analytic conductor: \(1.09794\)
Root analytic conductor: \(1.04782\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2200} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2200,\ (\ :0),\ 0.973 - 0.229i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9659961872\)
\(L(\frac12)\) \(\approx\) \(0.9659961872\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 \)
11 \( 1 + T \)
good3 \( 1 - iT^{2} \)
7 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
13 \( 1 + (-1.41 + 1.41i)T - iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - 2iT - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + 2iT - T^{2} \)
97 \( 1 + iT^{2} \)
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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

−9.058281687496977568454186937814, −8.412467579818979696560524854889, −8.073263207678984998309765517624, −7.46266435326599964896070341584, −5.87690958973760945538211806125, −5.31854627549538748859537638834, −4.45342676446941353135708098279, −3.06881886094146555453221317696, −2.37641978059221074765050535231, −1.42907058741349673759557092790, 0.992271787785488080269136914161, 1.88321145984844243076191973571, 3.73045533342023909348851020558, 4.46680788132998813583142046980, 5.30552429198928725267934640300, 6.39046705974206993904776236631, 6.92508059963593773265096474182, 7.76991527052345351417433032793, 8.325546287104669505089731806789, 9.041238007568505233611072470964

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