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Properties

Label 2-10e2-20.3-c3-0-6
Degree $2$
Conductor $100$
Sign $0.902 - 0.430i$
Analytic cond. $5.90019$
Root an. cond. $2.42903$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.933 − 2.66i)2-s + (−3.69 + 3.69i)3-s + (−6.25 − 4.98i)4-s + (6.41 + 13.3i)6-s + (19.3 + 19.3i)7-s + (−19.1 + 12.0i)8-s − 0.287i·9-s − 23.4i·11-s + (41.5 − 4.69i)12-s + (62.9 + 62.9i)13-s + (69.7 − 33.5i)14-s + (14.2 + 62.3i)16-s + (−0.872 + 0.872i)17-s + (−0.768 − 0.268i)18-s + 57.0·19-s + ⋯
L(s)  = 1  + (0.330 − 0.943i)2-s + (−0.710 + 0.710i)3-s + (−0.782 − 0.623i)4-s + (0.436 + 0.905i)6-s + (1.04 + 1.04i)7-s + (−0.846 + 0.532i)8-s − 0.0106i·9-s − 0.641i·11-s + (0.998 − 0.112i)12-s + (1.34 + 1.34i)13-s + (1.33 − 0.641i)14-s + (0.223 + 0.974i)16-s + (−0.0124 + 0.0124i)17-s + (−0.0100 − 0.00351i)18-s + 0.688·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 - 0.430i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.902 - 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $0.902 - 0.430i$
Analytic conductor: \(5.90019\)
Root analytic conductor: \(2.42903\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{100} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 100,\ (\ :3/2),\ 0.902 - 0.430i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.34044 + 0.303350i\)
\(L(\frac12)\) \(\approx\) \(1.34044 + 0.303350i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.933 + 2.66i)T \)
5 \( 1 \)
good3 \( 1 + (3.69 - 3.69i)T - 27iT^{2} \)
7 \( 1 + (-19.3 - 19.3i)T + 343iT^{2} \)
11 \( 1 + 23.4iT - 1.33e3T^{2} \)
13 \( 1 + (-62.9 - 62.9i)T + 2.19e3iT^{2} \)
17 \( 1 + (0.872 - 0.872i)T - 4.91e3iT^{2} \)
19 \( 1 - 57.0T + 6.85e3T^{2} \)
23 \( 1 + (116. - 116. i)T - 1.21e4iT^{2} \)
29 \( 1 - 121. iT - 2.43e4T^{2} \)
31 \( 1 + 66.8iT - 2.97e4T^{2} \)
37 \( 1 + (36.6 - 36.6i)T - 5.06e4iT^{2} \)
41 \( 1 + 302.T + 6.89e4T^{2} \)
43 \( 1 + (-190. + 190. i)T - 7.95e4iT^{2} \)
47 \( 1 + (-54.1 - 54.1i)T + 1.03e5iT^{2} \)
53 \( 1 + (36.5 + 36.5i)T + 1.48e5iT^{2} \)
59 \( 1 - 401.T + 2.05e5T^{2} \)
61 \( 1 - 509.T + 2.26e5T^{2} \)
67 \( 1 + (-187. - 187. i)T + 3.00e5iT^{2} \)
71 \( 1 + 584. iT - 3.57e5T^{2} \)
73 \( 1 + (436. + 436. i)T + 3.89e5iT^{2} \)
79 \( 1 - 608.T + 4.93e5T^{2} \)
83 \( 1 + (-124. + 124. i)T - 5.71e5iT^{2} \)
89 \( 1 + 684. iT - 7.04e5T^{2} \)
97 \( 1 + (-384. + 384. i)T - 9.12e5iT^{2} \)
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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

−13.50750678008892460567060103681, −11.75717669287684417470383698773, −11.57051223058542776165233434822, −10.62214798950823794158678791532, −9.312597403263430584906068812723, −8.376576018920409164831799487704, −5.93209540783476147373549615191, −5.11641453275015856797530918455, −3.82835872700854510836060039776, −1.77865971169596005255741123997, 0.858314055116101834916484838929, 3.93648308840266552925542615739, 5.34102277285598741678290958494, 6.46683092776233697297284181855, 7.54732072106689674100662714584, 8.358159214868836511093425016900, 10.16929437193553671922576321971, 11.40811699880104017423427223714, 12.50010756944029632892857956793, 13.38844551755893135659291540941

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