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Properties

Label 2-10e2-4.3-c10-0-82
Degree $2$
Conductor $100$
Sign $-0.939 + 0.343i$
Analytic cond. $63.5357$
Root an. cond. $7.97093$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (31.5 − 5.58i)2-s − 275. i·3-s + (961. − 351. i)4-s + (−1.53e3 − 8.68e3i)6-s + 1.93e4i·7-s + (2.83e4 − 1.64e4i)8-s − 1.69e4·9-s − 1.71e5i·11-s + (−9.70e4 − 2.65e5i)12-s − 2.03e5·13-s + (1.07e5 + 6.08e5i)14-s + (8.00e5 − 6.76e5i)16-s − 2.12e6·17-s + (−5.33e5 + 9.45e4i)18-s − 5.59e5i·19-s + ⋯
L(s)  = 1  + (0.984 − 0.174i)2-s − 1.13i·3-s + (0.939 − 0.343i)4-s + (−0.197 − 1.11i)6-s + 1.14i·7-s + (0.864 − 0.502i)8-s − 0.286·9-s − 1.06i·11-s + (−0.389 − 1.06i)12-s − 0.547·13-s + (0.200 + 1.13i)14-s + (0.763 − 0.645i)16-s − 1.49·17-s + (−0.282 + 0.0500i)18-s − 0.225i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.343i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.939 + 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $-0.939 + 0.343i$
Analytic conductor: \(63.5357\)
Root analytic conductor: \(7.97093\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{100} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 100,\ (\ :5),\ -0.939 + 0.343i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.541934 - 3.05733i\)
\(L(\frac12)\) \(\approx\) \(0.541934 - 3.05733i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-31.5 + 5.58i)T \)
5 \( 1 \)
good3 \( 1 + 275. iT - 5.90e4T^{2} \)
7 \( 1 - 1.93e4iT - 2.82e8T^{2} \)
11 \( 1 + 1.71e5iT - 2.59e10T^{2} \)
13 \( 1 + 2.03e5T + 1.37e11T^{2} \)
17 \( 1 + 2.12e6T + 2.01e12T^{2} \)
19 \( 1 + 5.59e5iT - 6.13e12T^{2} \)
23 \( 1 + 8.97e6iT - 4.14e13T^{2} \)
29 \( 1 + 6.60e6T + 4.20e14T^{2} \)
31 \( 1 + 4.19e7iT - 8.19e14T^{2} \)
37 \( 1 + 3.19e7T + 4.80e15T^{2} \)
41 \( 1 - 1.21e8T + 1.34e16T^{2} \)
43 \( 1 + 1.35e8iT - 2.16e16T^{2} \)
47 \( 1 - 3.08e8iT - 5.25e16T^{2} \)
53 \( 1 + 7.27e8T + 1.74e17T^{2} \)
59 \( 1 - 2.51e8iT - 5.11e17T^{2} \)
61 \( 1 + 9.40e8T + 7.13e17T^{2} \)
67 \( 1 - 2.07e9iT - 1.82e18T^{2} \)
71 \( 1 + 1.90e9iT - 3.25e18T^{2} \)
73 \( 1 - 1.72e9T + 4.29e18T^{2} \)
79 \( 1 - 2.36e9iT - 9.46e18T^{2} \)
83 \( 1 + 5.22e9iT - 1.55e19T^{2} \)
89 \( 1 - 8.10e9T + 3.11e19T^{2} \)
97 \( 1 - 5.70e9T + 7.37e19T^{2} \)
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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

−11.74625798174059690136639006181, −10.82625372310815903775669806070, −9.088105859731771764091399916805, −7.82137679121792477190418843618, −6.56900832608531096426232236653, −5.88295036925802328738892398301, −4.47360378108160995187077217323, −2.71703247389502986655538762842, −2.02336042608752619505830749707, −0.48433095996616068503541650723, 1.76861502029858739907536005493, 3.43160139409064096910033214779, 4.37502984030029991927896668145, 5.02045562378178888594327814927, 6.73673124143035088257387173332, 7.58900156247357046886223357509, 9.400408632594433213488726966932, 10.41195630885598500267172450216, 11.15681301093511538572877304891, 12.48629450831648406776497489840

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