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Properties

Label 1-67-67.53-r1-0-0
Degree $1$
Conductor $67$
Sign $-0.903 - 0.428i$
Analytic cond. $7.20014$
Root an. cond. $7.20014$
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  + (0.654 − 0.755i)2-s + (−0.415 − 0.909i)3-s + (−0.142 − 0.989i)4-s + (0.959 − 0.281i)5-s + (−0.959 − 0.281i)6-s + (0.654 − 0.755i)7-s + (−0.841 − 0.540i)8-s + (−0.654 + 0.755i)9-s + (0.415 − 0.909i)10-s + (0.959 − 0.281i)11-s + (−0.841 + 0.540i)12-s + (−0.841 + 0.540i)13-s + (−0.142 − 0.989i)14-s + (−0.654 − 0.755i)15-s + (−0.959 + 0.281i)16-s + (−0.142 + 0.989i)17-s + ⋯
L(s)  = 1  + (0.654 − 0.755i)2-s + (−0.415 − 0.909i)3-s + (−0.142 − 0.989i)4-s + (0.959 − 0.281i)5-s + (−0.959 − 0.281i)6-s + (0.654 − 0.755i)7-s + (−0.841 − 0.540i)8-s + (−0.654 + 0.755i)9-s + (0.415 − 0.909i)10-s + (0.959 − 0.281i)11-s + (−0.841 + 0.540i)12-s + (−0.841 + 0.540i)13-s + (−0.142 − 0.989i)14-s + (−0.654 − 0.755i)15-s + (−0.959 + 0.281i)16-s + (−0.142 + 0.989i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.903 - 0.428i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.903 - 0.428i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(67\)
Sign: $-0.903 - 0.428i$
Analytic conductor: \(7.20014\)
Root analytic conductor: \(7.20014\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{67} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 67,\ (1:\ ),\ -0.903 - 0.428i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4892271064 - 2.173754521i\)
\(L(\frac12)\) \(\approx\) \(0.4892271064 - 2.173754521i\)
\(L(1)\) \(\approx\) \(0.9757341412 - 1.191489898i\)
\(L(1)\) \(\approx\) \(0.9757341412 - 1.191489898i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad67 \( 1 \)
good2 \( 1 + (0.654 - 0.755i)T \)
3 \( 1 + (-0.415 - 0.909i)T \)
5 \( 1 + (0.959 - 0.281i)T \)
7 \( 1 + (0.654 - 0.755i)T \)
11 \( 1 + (0.959 - 0.281i)T \)
13 \( 1 + (-0.841 + 0.540i)T \)
17 \( 1 + (-0.142 + 0.989i)T \)
19 \( 1 + (-0.654 - 0.755i)T \)
23 \( 1 + (0.415 + 0.909i)T \)
29 \( 1 + T \)
31 \( 1 + (-0.841 - 0.540i)T \)
37 \( 1 + T \)
41 \( 1 + (0.142 - 0.989i)T \)
43 \( 1 + (0.142 - 0.989i)T \)
47 \( 1 + (0.415 + 0.909i)T \)
53 \( 1 + (0.142 + 0.989i)T \)
59 \( 1 + (0.841 + 0.540i)T \)
61 \( 1 + (0.959 + 0.281i)T \)
71 \( 1 + (-0.142 - 0.989i)T \)
73 \( 1 + (-0.959 - 0.281i)T \)
79 \( 1 + (-0.841 + 0.540i)T \)
83 \( 1 + (-0.959 + 0.281i)T \)
89 \( 1 + (0.415 - 0.909i)T \)
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

−32.45716603563462343050579485923, −31.462036470407930478012226070606, −30.1483285492744245378590602497, −29.02950454389454958295809676744, −27.52047733492426598300972973480, −26.76742853816583338372525706735, −25.246395858787299032796248783847, −24.823258435330751086627508476696, −23.08764295044750956457881751767, −22.14523131608543765898101918656, −21.53309880415113216079202391308, −20.46965551252696208886034476278, −18.10173452079142720791796737035, −17.312157375117300097259094340533, −16.28230077707637428035454005393, −14.706940310326571044983994308453, −14.53313973286548084027809144141, −12.609745905253911236559683057399, −11.44569108686949086942190036336, −9.80287428529523709942898426854, −8.63333756333185562583096619800, −6.67928947542098951361700975290, −5.52955636942989397525429881015, −4.56574090835911586405568039972, −2.7149539546885909365695552442, 1.07754618106825380392565727177, 2.17652015839765108608257371865, 4.40308678640736028450713040806, 5.76086065307963205921268056392, 6.94840690155216206500947661524, 8.98904502678760525331534245803, 10.53943416312599974236421288582, 11.60613914662326662208536106204, 12.81541568219909224446924026507, 13.76766033824134628817337366251, 14.59336434716889384256663567575, 16.99527349309525461751425123000, 17.624812235962011309992859456779, 19.155873725784678403634622596777, 19.998101603211749710777698984276, 21.43746345365943805144021079648, 22.19032481153384322152993894572, 23.72114550914326186726364866021, 24.19124211470299774690695448416, 25.40688950609650075704326061663, 27.26185565678658409738021626823, 28.43433043914894156473135835641, 29.40210005497917157618369530989, 30.01873129058858299304832428037, 30.90693774767611239963871639358

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