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Properties

Label 2-10-5.3-c6-0-1
Degree $2$
Conductor $10$
Sign $-0.640 + 0.767i$
Analytic cond. $2.30054$
Root an. cond. $1.51675$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4 + 4i)2-s + (−23 − 23i)3-s − 32i·4-s + (−75 − 100i)5-s + 184·6-s + (−247 + 247i)7-s + (128 + 128i)8-s + 329i·9-s + (700 + 100i)10-s + 1.40e3·11-s + (−736 + 736i)12-s + (−2.70e3 − 2.70e3i)13-s − 1.97e3i·14-s + (−575 + 4.02e3i)15-s − 1.02e3·16-s + (2.59e3 − 2.59e3i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (−0.851 − 0.851i)3-s − 0.5i·4-s + (−0.599 − 0.800i)5-s + 0.851·6-s + (−0.720 + 0.720i)7-s + (0.250 + 0.250i)8-s + 0.451i·9-s + (0.700 + 0.100i)10-s + 1.05·11-s + (−0.425 + 0.425i)12-s + (−1.23 − 1.23i)13-s − 0.720i·14-s + (−0.170 + 1.19i)15-s − 0.250·16-s + (0.527 − 0.527i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.640 + 0.767i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.640 + 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $-0.640 + 0.767i$
Analytic conductor: \(2.30054\)
Root analytic conductor: \(1.51675\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{10} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :3),\ -0.640 + 0.767i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.173702 - 0.371216i\)
\(L(\frac12)\) \(\approx\) \(0.173702 - 0.371216i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (4 - 4i)T \)
5 \( 1 + (75 + 100i)T \)
good3 \( 1 + (23 + 23i)T + 729iT^{2} \)
7 \( 1 + (247 - 247i)T - 1.17e5iT^{2} \)
11 \( 1 - 1.40e3T + 1.77e6T^{2} \)
13 \( 1 + (2.70e3 + 2.70e3i)T + 4.82e6iT^{2} \)
17 \( 1 + (-2.59e3 + 2.59e3i)T - 2.41e7iT^{2} \)
19 \( 1 + 1.72e3iT - 4.70e7T^{2} \)
23 \( 1 + (-2.13e3 - 2.13e3i)T + 1.48e8iT^{2} \)
29 \( 1 + 3.05e4iT - 5.94e8T^{2} \)
31 \( 1 + 3.78e4T + 8.87e8T^{2} \)
37 \( 1 + (-3.71e4 + 3.71e4i)T - 2.56e9iT^{2} \)
41 \( 1 + 3.54e4T + 4.75e9T^{2} \)
43 \( 1 + (-3.91e4 - 3.91e4i)T + 6.32e9iT^{2} \)
47 \( 1 + (-9.51e4 + 9.51e4i)T - 1.07e10iT^{2} \)
53 \( 1 + (-3.60e4 - 3.60e4i)T + 2.21e10iT^{2} \)
59 \( 1 - 3.59e4iT - 4.21e10T^{2} \)
61 \( 1 - 8.33e4T + 5.15e10T^{2} \)
67 \( 1 + (-6.08e4 + 6.08e4i)T - 9.04e10iT^{2} \)
71 \( 1 + 4.03e4T + 1.28e11T^{2} \)
73 \( 1 + (1.29e5 + 1.29e5i)T + 1.51e11iT^{2} \)
79 \( 1 + 5.24e5iT - 2.43e11T^{2} \)
83 \( 1 + (1.14e5 + 1.14e5i)T + 3.26e11iT^{2} \)
89 \( 1 + 1.87e5iT - 4.96e11T^{2} \)
97 \( 1 + (-5.32e5 + 5.32e5i)T - 8.32e11iT^{2} \)
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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

−18.97066514379148405079858323693, −17.57066677305625424790176005393, −16.59774190339917888794105022191, −15.14307447814799684737386093760, −12.75135291853885834326910510800, −11.80748732035280005999843675040, −9.348948698564673596768843027687, −7.41579948042753569680580278597, −5.67199416372430237408373390717, −0.45719974308732778039320908703, 3.97447928207397225156487965858, 6.96872610397053227181765433980, 9.680538275033016722032587633368, 10.84991820920565446614813894006, 12.06791482750842796238168573781, 14.51473847103721798149995301957, 16.37103147277319680501923015314, 17.05338442944964612721408766329, 18.95270406139479278822887060021, 19.89905126297752912454729502298

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