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Properties

Label 1-805-805.597-r0-0-0
Degree $1$
Conductor $805$
Sign $0.578 - 0.815i$
Analytic cond. $3.73840$
Root an. cond. $3.73840$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)2-s + (0.866 + 0.5i)3-s + (0.5 − 0.866i)4-s + 6-s − i·8-s + (0.5 + 0.866i)9-s + (0.5 − 0.866i)11-s + (0.866 − 0.5i)12-s − i·13-s + (−0.5 − 0.866i)16-s + (0.866 + 0.5i)17-s + (0.866 + 0.5i)18-s + (−0.5 − 0.866i)19-s − i·22-s + (0.5 − 0.866i)24-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.866 + 0.5i)3-s + (0.5 − 0.866i)4-s + 6-s − i·8-s + (0.5 + 0.866i)9-s + (0.5 − 0.866i)11-s + (0.866 − 0.5i)12-s − i·13-s + (−0.5 − 0.866i)16-s + (0.866 + 0.5i)17-s + (0.866 + 0.5i)18-s + (−0.5 − 0.866i)19-s − i·22-s + (0.5 − 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.578 - 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.578 - 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(805\)    =    \(5 \cdot 7 \cdot 23\)
Sign: $0.578 - 0.815i$
Analytic conductor: \(3.73840\)
Root analytic conductor: \(3.73840\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{805} (597, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 805,\ (0:\ ),\ 0.578 - 0.815i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.039277779 - 1.570414875i\)
\(L(\frac12)\) \(\approx\) \(3.039277779 - 1.570414875i\)
\(L(1)\) \(\approx\) \(2.194409132 - 0.6583293809i\)
\(L(1)\) \(\approx\) \(2.194409132 - 0.6583293809i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 - iT \)
17 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 - T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.866 + 0.5i)T \)
41 \( 1 + T \)
43 \( 1 + iT \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (-0.866 - 0.5i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 + T \)
73 \( 1 + (0.866 + 0.5i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + iT \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 - iT \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.55112405583281218564519198333, −21.50148668008161013925831691013, −20.7508951781446165856045335150, −20.27942650214855659521421393908, −19.17038256357541624468239838776, −18.49992269171374808365714601174, −17.361253448817803110835300674691, −16.65982372062319935181803831433, −15.67621352779217373670115820932, −14.79359916460945429221943834637, −14.29850837079183388878079627802, −13.65078086776397101708912703423, −12.50062322581027709495968092518, −12.26114769488177997289116741809, −11.135706865162203717007368709227, −9.6897292593820276440310720320, −8.972265949949343903759789738585, −7.86227004352516132872442982293, −7.261689015205050699019399602825, −6.49225858000880910855078525655, −5.46842571053247718143591607971, −4.16299342296535817018021261988, −3.70080000761736411938145266551, −2.39828603421980123810190292122, −1.67615315384955461478588662875, 1.1534294368601825673364070647, 2.36188792563090551800235038388, 3.30927370880220119207494081440, 3.82309514798553200282777773084, 4.99232743113073931668299371685, 5.75089273034464121610470418406, 6.90989522264517095918090290118, 8.01421422616262542219572666150, 8.94467359133966871416522292827, 9.86427789082175202022745746030, 10.68128864386015318446469189696, 11.32933619544945940680911960075, 12.61203873598469625211294545262, 13.14162023067116035564252540088, 14.11682638770428596798007438502, 14.62957336160180215544182276649, 15.44371032090723106531595627347, 16.13949665440974934694415022415, 17.14122213241793236008036749095, 18.51033697231230183605740758593, 19.320884632163827206925960006213, 19.80113092532588267343713811478, 20.65834690676164545464257506813, 21.333431169300971433498564635690, 21.98226668512663887312032522798

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