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Properties

Label 2-10e2-20.7-c3-0-19
Degree $2$
Conductor $100$
Sign $0.608 + 0.793i$
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  + (2.78 + 0.510i)2-s + (−4.02 − 4.02i)3-s + (7.47 + 2.83i)4-s + (−9.15 − 13.2i)6-s + (14.4 − 14.4i)7-s + (19.3 + 11.7i)8-s + 5.46i·9-s − 47.0i·11-s + (−18.6 − 41.5i)12-s + (8.79 − 8.79i)13-s + (47.5 − 32.8i)14-s + (47.8 + 42.4i)16-s + (26.4 + 26.4i)17-s + (−2.79 + 15.2i)18-s − 49.8·19-s + ⋯
L(s)  = 1  + (0.983 + 0.180i)2-s + (−0.775 − 0.775i)3-s + (0.934 + 0.354i)4-s + (−0.622 − 0.902i)6-s + (0.779 − 0.779i)7-s + (0.855 + 0.517i)8-s + 0.202i·9-s − 1.28i·11-s + (−0.449 − 1.00i)12-s + (0.187 − 0.187i)13-s + (0.907 − 0.626i)14-s + (0.747 + 0.663i)16-s + (0.377 + 0.377i)17-s + (−0.0365 + 0.199i)18-s − 0.601·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.608 + 0.793i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.608 + 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $0.608 + 0.793i$
Analytic conductor: \(5.90019\)
Root analytic conductor: \(2.42903\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{100} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 100,\ (\ :3/2),\ 0.608 + 0.793i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.11509 - 1.04328i\)
\(L(\frac12)\) \(\approx\) \(2.11509 - 1.04328i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.78 - 0.510i)T \)
5 \( 1 \)
good3 \( 1 + (4.02 + 4.02i)T + 27iT^{2} \)
7 \( 1 + (-14.4 + 14.4i)T - 343iT^{2} \)
11 \( 1 + 47.0iT - 1.33e3T^{2} \)
13 \( 1 + (-8.79 + 8.79i)T - 2.19e3iT^{2} \)
17 \( 1 + (-26.4 - 26.4i)T + 4.91e3iT^{2} \)
19 \( 1 + 49.8T + 6.85e3T^{2} \)
23 \( 1 + (-41.2 - 41.2i)T + 1.21e4iT^{2} \)
29 \( 1 - 247. iT - 2.43e4T^{2} \)
31 \( 1 - 62.3iT - 2.97e4T^{2} \)
37 \( 1 + (-73.2 - 73.2i)T + 5.06e4iT^{2} \)
41 \( 1 - 118.T + 6.89e4T^{2} \)
43 \( 1 + (245. + 245. i)T + 7.95e4iT^{2} \)
47 \( 1 + (125. - 125. i)T - 1.03e5iT^{2} \)
53 \( 1 + (-326. + 326. i)T - 1.48e5iT^{2} \)
59 \( 1 + 365.T + 2.05e5T^{2} \)
61 \( 1 + 268.T + 2.26e5T^{2} \)
67 \( 1 + (-112. + 112. i)T - 3.00e5iT^{2} \)
71 \( 1 - 559. iT - 3.57e5T^{2} \)
73 \( 1 + (215. - 215. i)T - 3.89e5iT^{2} \)
79 \( 1 - 1.17e3T + 4.93e5T^{2} \)
83 \( 1 + (-592. - 592. i)T + 5.71e5iT^{2} \)
89 \( 1 + 552. iT - 7.04e5T^{2} \)
97 \( 1 + (460. + 460. i)T + 9.12e5iT^{2} \)
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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

−13.22771961510301362504506373309, −12.31371050553294535903063480737, −11.26016303109392774098406783961, −10.71376572424348344554437503356, −8.387314415714306437826830226303, −7.24351620251863888801036307040, −6.22496427739086583395637681296, −5.16196423215554074952201854857, −3.54767598200911756270002064627, −1.26524440151018174477050148629, 2.19269831902665726336804065097, 4.32790778046088410210228404574, 5.06780036656488058429900597422, 6.20902742506405040916976865959, 7.76594327166214337405132997420, 9.600742884734268260999696110892, 10.64324654455843677933431913167, 11.57957567808578348621061463138, 12.25324729065119892705034176560, 13.49607473364385934505195315010

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