Properties

Degree 2
Conductor $ 2 \cdot 5 $
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

\( d \)  =  \(2\)
\( N \)  =  \(10\)    =    \(2 \cdot 5\)
\( \varepsilon \)  =  $0.973 + 0.229i$
motivic weight  =  \(2\)
character  :  $\chi_{10} (7, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 10,\ (\ :1),\ 0.973 + 0.229i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.485972 - 0.0565836i\)
\(L(\frac12)\)  \(\approx\)  \(0.485972 - 0.0565836i\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 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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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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