Properties

Label 2-10e2-20.3-c3-0-2
Degree $2$
Conductor $100$
Sign $-0.999 + 0.0185i$
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  + (−2.66 + 0.933i)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 + (−12.0 + 19.1i)8-s − 0.287i·9-s + 23.4i·11-s + (−4.69 + 41.5i)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 + (0.268 + 0.768i)18-s − 57.0·19-s + ⋯
L(s)  = 1  + (−0.943 + 0.330i)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 + (−0.532 + 0.846i)8-s − 0.0106i·9-s + 0.641i·11-s + (−0.112 + 0.998i)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 + (0.00351 + 0.0100i)18-s − 0.688·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0185i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0185i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $-0.999 + 0.0185i$
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: \((2,\ 100,\ (\ :3/2),\ -0.999 + 0.0185i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.00437468 - 0.472028i\)
\(L(\frac12)\) \(\approx\) \(0.00437468 - 0.472028i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.66 - 0.933i)T \)
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

−14.45745103819044231137434932491, −12.39828638698897946266493203374, −11.55880290032207310933026463699, −10.51380993890187460371039589476, −9.764128217121988856117337206339, −8.424113398151205012364050754739, −7.45331631279708008575024765328, −5.65439997333687831548906347586, −4.99703770796657868972886924781, −2.17758082336768653725211242440, 0.37866571276377995695018880138, 1.91456050823536014012144313007, 4.30764699513120455125892631575, 6.36021795638152970970252436772, 7.28121247079804753799039581943, 8.286110123872748800252744954997, 9.717327674652725007814823438412, 10.88133427725649815557158082052, 11.65593287169540832353507186289, 12.40714409616324157188509335756

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