Properties

Label 2-10e2-25.19-c3-0-7
Degree $2$
Conductor $100$
Sign $-0.999 - 0.00827i$
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  + (−5.78 − 7.95i)3-s + (10.1 − 4.71i)5-s − 24.7i·7-s + (−21.5 + 66.3i)9-s + (1.97 + 6.07i)11-s + (−62.0 − 20.1i)13-s + (−96.1 − 53.3i)15-s + (−35.7 + 49.2i)17-s + (43.8 + 31.8i)19-s + (−196. + 143. i)21-s + (7.45 − 2.42i)23-s + (80.4 − 95.6i)25-s + (400. − 130. i)27-s + (105. − 76.4i)29-s + (−231. − 168. i)31-s + ⋯
L(s)  = 1  + (−1.11 − 1.53i)3-s + (0.906 − 0.422i)5-s − 1.33i·7-s + (−0.798 + 2.45i)9-s + (0.0540 + 0.166i)11-s + (−1.32 − 0.429i)13-s + (−1.65 − 0.918i)15-s + (−0.510 + 0.702i)17-s + (0.529 + 0.384i)19-s + (−2.04 + 1.48i)21-s + (0.0675 − 0.0219i)23-s + (0.643 − 0.765i)25-s + (2.85 − 0.926i)27-s + (0.673 − 0.489i)29-s + (−1.34 − 0.975i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00827i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00827i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $-0.999 - 0.00827i$
Analytic conductor: \(5.90019\)
Root analytic conductor: \(2.42903\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{100} (69, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 100,\ (\ :3/2),\ -0.999 - 0.00827i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.00358217 + 0.865829i\)
\(L(\frac12)\) \(\approx\) \(0.00358217 + 0.865829i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-10.1 + 4.71i)T \)
good3 \( 1 + (5.78 + 7.95i)T + (-8.34 + 25.6i)T^{2} \)
7 \( 1 + 24.7iT - 343T^{2} \)
11 \( 1 + (-1.97 - 6.07i)T + (-1.07e3 + 782. i)T^{2} \)
13 \( 1 + (62.0 + 20.1i)T + (1.77e3 + 1.29e3i)T^{2} \)
17 \( 1 + (35.7 - 49.2i)T + (-1.51e3 - 4.67e3i)T^{2} \)
19 \( 1 + (-43.8 - 31.8i)T + (2.11e3 + 6.52e3i)T^{2} \)
23 \( 1 + (-7.45 + 2.42i)T + (9.84e3 - 7.15e3i)T^{2} \)
29 \( 1 + (-105. + 76.4i)T + (7.53e3 - 2.31e4i)T^{2} \)
31 \( 1 + (231. + 168. i)T + (9.20e3 + 2.83e4i)T^{2} \)
37 \( 1 + (-237. - 77.1i)T + (4.09e4 + 2.97e4i)T^{2} \)
41 \( 1 + (-18.4 + 56.7i)T + (-5.57e4 - 4.05e4i)T^{2} \)
43 \( 1 - 32.9iT - 7.95e4T^{2} \)
47 \( 1 + (339. + 467. i)T + (-3.20e4 + 9.87e4i)T^{2} \)
53 \( 1 + (0.813 + 1.11i)T + (-4.60e4 + 1.41e5i)T^{2} \)
59 \( 1 + (19.0 - 58.6i)T + (-1.66e5 - 1.20e5i)T^{2} \)
61 \( 1 + (185. + 571. i)T + (-1.83e5 + 1.33e5i)T^{2} \)
67 \( 1 + (357. - 491. i)T + (-9.29e4 - 2.86e5i)T^{2} \)
71 \( 1 + (-544. + 395. i)T + (1.10e5 - 3.40e5i)T^{2} \)
73 \( 1 + (-331. + 107. i)T + (3.14e5 - 2.28e5i)T^{2} \)
79 \( 1 + (7.93 - 5.76i)T + (1.52e5 - 4.68e5i)T^{2} \)
83 \( 1 + (-719. + 990. i)T + (-1.76e5 - 5.43e5i)T^{2} \)
89 \( 1 + (248. + 764. i)T + (-5.70e5 + 4.14e5i)T^{2} \)
97 \( 1 + (-528. - 726. i)T + (-2.82e5 + 8.68e5i)T^{2} \)
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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

−12.97239635541963406924542460467, −12.02420510522258372336070521445, −10.83054748524242106252606183979, −9.886237375559238178270797086702, −7.942099909182712165271121378382, −7.07367601142801113915338684190, −6.04158262123678028616792768044, −4.85813022442964141767000245041, −1.95198934464948595469441645140, −0.55016379134604358834996402531, 2.82981163853603424146144401814, 4.84019788982369933543693811866, 5.54467792075375987243540460625, 6.68268346596412712633285692325, 9.222367626364464057049959457307, 9.490468244874614592428630239010, 10.72470267189558768654103125295, 11.59338495161403871408473909512, 12.50911643709753817043803495272, 14.31609741393642306433375694550

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