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Properties

Label 1-65-65.38-r1-0-0
Degree $1$
Conductor $65$
Sign $-0.525 - 0.850i$
Analytic cond. $6.98522$
Root an. cond. $6.98522$
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  + i·2-s + i·3-s − 4-s − 6-s + i·7-s − i·8-s − 9-s − 11-s − i·12-s − 14-s + 16-s − i·17-s − i·18-s + 19-s − 21-s − i·22-s + ⋯
L(s)  = 1  + i·2-s + i·3-s − 4-s − 6-s + i·7-s − i·8-s − 9-s − 11-s − i·12-s − 14-s + 16-s − i·17-s − i·18-s + 19-s − 21-s − i·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $-0.525 - 0.850i$
Analytic conductor: \(6.98522\)
Root analytic conductor: \(6.98522\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 65,\ (1:\ ),\ -0.525 - 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3695570296 + 0.6628391488i\)
\(L(\frac12)\) \(\approx\) \(-0.3695570296 + 0.6628391488i\)
\(L(1)\) \(\approx\) \(0.4097197214 + 0.6629404352i\)
\(L(1)\) \(\approx\) \(0.4097197214 + 0.6629404352i\)

Euler product

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

−30.807790262876180284086190038120, −30.121622106617950563338806991614, −29.080336009712332549022597510405, −28.36721855947621313906891060043, −26.76311938321296310671869219267, −25.92642370030320721848675810338, −24.12635177655701527792065644199, −23.38778903570555983678039375267, −22.29874431025155860588918315258, −20.71456924587264684913341939995, −19.95255066224156165858986553104, −18.78907940802136336618600239613, −17.8970685259853354734479554077, −16.74675217499687804785617360013, −14.51577844851933707436307334913, −13.3949005881401251119789296790, −12.66404959536644353718705689311, −11.25948354792155985136902641183, −10.215960135529180046592771840994, −8.485565418463994628191552516707, −7.305271029617953847303244347751, −5.41552356075866429895562411336, −3.58367883989269684515199529165, −1.9703836689085405062464187484, −0.39184480300428952573422920640, 3.16650970393109837971195712624, 4.96461296713060060082698260418, 5.71608039558275707933346658337, 7.63028702906795903558473536279, 8.974308796279650592665140232656, 9.86861047716615103129680768, 11.589312047849249325603612442422, 13.29781052922602675923375034575, 14.64140163333971153728065132301, 15.63211386460687500551000774696, 16.26659549390168773048053462531, 17.73797521296180902044128825932, 18.740732273854421293077363214729, 20.53494960512183441830649917277, 21.76846758517232917750076188165, 22.54536526221187387989064245710, 23.7953321609730695738799987006, 25.11397479204308615333878166747, 25.969597850321002103977668614028, 27.01947603854789218692836183596, 27.92851310381020936779092254528, 28.95803419094287913273993086583, 31.24171123473786650650188874670, 31.56944923249218574601136857586, 32.76421617972623729726676073403

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