Properties

Degree $2$
Conductor $10$
Sign $0.973 - 0.229i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + i)2-s + (−2 − 2i)3-s − 2i·4-s + 5i·5-s + 4·6-s + (2 − 2i)7-s + (2 + 2i)8-s i·9-s + (−5 − 5i)10-s − 8·11-s + (−4 + 4i)12-s + (3 + 3i)13-s + 4i·14-s + (10 − 10i)15-s − 4·16-s + (7 − 7i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (−0.666 − 0.666i)3-s − 0.5i·4-s + i·5-s + 0.666·6-s + (0.285 − 0.285i)7-s + (0.250 + 0.250i)8-s − 0.111i·9-s + (−0.5 − 0.5i)10-s − 0.727·11-s + (−0.333 + 0.333i)12-s + (0.230 + 0.230i)13-s + 0.285i·14-s + (0.666 − 0.666i)15-s − 0.250·16-s + (0.411 − 0.411i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $0.973 - 0.229i$
Motivic weight: \(2\)
Character: $\chi_{10} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :1),\ 0.973 - 0.229i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.485972 + 0.0565836i\)
\(L(\frac12)\) \(\approx\) \(0.485972 + 0.0565836i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - i)T \)
5 \( 1 - 5iT \)
good3 \( 1 + (2 + 2i)T + 9iT^{2} \)
7 \( 1 + (-2 + 2i)T - 49iT^{2} \)
11 \( 1 + 8T + 121T^{2} \)
13 \( 1 + (-3 - 3i)T + 169iT^{2} \)
17 \( 1 + (-7 + 7i)T - 289iT^{2} \)
19 \( 1 - 20iT - 361T^{2} \)
23 \( 1 + (2 + 2i)T + 529iT^{2} \)
29 \( 1 + 40iT - 841T^{2} \)
31 \( 1 - 52T + 961T^{2} \)
37 \( 1 + (3 - 3i)T - 1.36e3iT^{2} \)
41 \( 1 + 8T + 1.68e3T^{2} \)
43 \( 1 + (42 + 42i)T + 1.84e3iT^{2} \)
47 \( 1 + (18 - 18i)T - 2.20e3iT^{2} \)
53 \( 1 + (-53 - 53i)T + 2.80e3iT^{2} \)
59 \( 1 - 20iT - 3.48e3T^{2} \)
61 \( 1 + 48T + 3.72e3T^{2} \)
67 \( 1 + (-62 + 62i)T - 4.48e3iT^{2} \)
71 \( 1 + 28T + 5.04e3T^{2} \)
73 \( 1 + (47 + 47i)T + 5.32e3iT^{2} \)
79 \( 1 - 6.24e3T^{2} \)
83 \( 1 + (-18 - 18i)T + 6.88e3iT^{2} \)
89 \( 1 + 80iT - 7.92e3T^{2} \)
97 \( 1 + (63 - 63i)T - 9.40e3iT^{2} \)
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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

−20.91840880760687132371699108974, −18.93596683106944575370378584993, −18.20751455061888268560137839128, −17.06911658410946393747976120432, −15.36823414016552186873536868229, −13.84524531625142941910940971889, −11.76761621055593255088125467995, −10.23281628919080119416287605008, −7.67558497356483516574264525363, −6.19183269878294484670114036539, 5.03540251964772219755791684917, 8.393364582816363596048703883492, 10.14654137527546697435584467874, 11.58377096028888216433546983152, 13.13024352722512341574554202219, 15.66615912972948546686921550101, 16.73674996682997615882681407582, 17.92223562712510451121164587598, 19.62802241243125925268870039984, 20.97670027424001189851460753414

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