Properties

Degree $2$
Conductor $10$
Sign $0.178 - 0.983i$
Motivic weight $5$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4i·2-s + 14i·3-s − 16·4-s + (55 + 10i)5-s − 56·6-s − 158i·7-s − 64i·8-s + 47·9-s + (−40 + 220i)10-s − 148·11-s − 224i·12-s + 684i·13-s + 632·14-s + (−140 + 770i)15-s + 256·16-s − 2.04e3i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.898i·3-s − 0.5·4-s + (0.983 + 0.178i)5-s − 0.635·6-s − 1.21i·7-s − 0.353i·8-s + 0.193·9-s + (−0.126 + 0.695i)10-s − 0.368·11-s − 0.449i·12-s + 1.12i·13-s + 0.861·14-s + (−0.160 + 0.883i)15-s + 0.250·16-s − 1.71i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.178 - 0.983i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.178 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $0.178 - 0.983i$
Motivic weight: \(5\)
Character: $\chi_{10} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :5/2),\ 0.178 - 0.983i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.950751 + 0.793474i\)
\(L(\frac12)\) \(\approx\) \(0.950751 + 0.793474i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 4iT \)
5 \( 1 + (-55 - 10i)T \)
good3 \( 1 - 14iT - 243T^{2} \)
7 \( 1 + 158iT - 1.68e4T^{2} \)
11 \( 1 + 148T + 1.61e5T^{2} \)
13 \( 1 - 684iT - 3.71e5T^{2} \)
17 \( 1 + 2.04e3iT - 1.41e6T^{2} \)
19 \( 1 + 2.22e3T + 2.47e6T^{2} \)
23 \( 1 + 1.24e3iT - 6.43e6T^{2} \)
29 \( 1 - 270T + 2.05e7T^{2} \)
31 \( 1 + 2.04e3T + 2.86e7T^{2} \)
37 \( 1 - 4.37e3iT - 6.93e7T^{2} \)
41 \( 1 + 2.39e3T + 1.15e8T^{2} \)
43 \( 1 - 2.29e3iT - 1.47e8T^{2} \)
47 \( 1 - 1.06e4iT - 2.29e8T^{2} \)
53 \( 1 - 2.96e3iT - 4.18e8T^{2} \)
59 \( 1 - 3.97e4T + 7.14e8T^{2} \)
61 \( 1 + 4.22e4T + 8.44e8T^{2} \)
67 \( 1 + 3.20e4iT - 1.35e9T^{2} \)
71 \( 1 + 4.24e3T + 1.80e9T^{2} \)
73 \( 1 - 3.01e4iT - 2.07e9T^{2} \)
79 \( 1 + 3.52e4T + 3.07e9T^{2} \)
83 \( 1 + 2.78e4iT - 3.93e9T^{2} \)
89 \( 1 - 8.52e4T + 5.58e9T^{2} \)
97 \( 1 - 9.72e4iT - 8.58e9T^{2} \)
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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

−20.63631268222401779868978150617, −18.56355373914592847949389886251, −17.05850236308791270866577775569, −16.18098083914274725706747896861, −14.50472295777770541541737003745, −13.40296404236440758697169464654, −10.57890465133649213233181899428, −9.379264379403550435948853783827, −6.89034138045290143199524724162, −4.58996945625990890057076318897, 2.02947078065342118326702508644, 5.86300359197973539959649504789, 8.514372490718778541168876741302, 10.36555378505865412595578756940, 12.52318821463482741978161202808, 13.11028404795236176114711471885, 15.05619563843942359927211657003, 17.42858007770352505493729629267, 18.35908324857109319320129519694, 19.43513243283377244668263818845

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