Properties

Degree $1$
Conductor $67$
Sign $0.0327 - 0.999i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.415 − 0.909i)2-s + (0.841 − 0.540i)3-s + (−0.654 − 0.755i)4-s + (−0.142 + 0.989i)5-s + (−0.142 − 0.989i)6-s + (0.415 − 0.909i)7-s + (−0.959 + 0.281i)8-s + (0.415 − 0.909i)9-s + (0.841 + 0.540i)10-s + (−0.142 + 0.989i)11-s + (−0.959 − 0.281i)12-s + (−0.959 − 0.281i)13-s + (−0.654 − 0.755i)14-s + (0.415 + 0.909i)15-s + (−0.142 + 0.989i)16-s + (−0.654 + 0.755i)17-s + ⋯
L(s,χ)  = 1  + (0.415 − 0.909i)2-s + (0.841 − 0.540i)3-s + (−0.654 − 0.755i)4-s + (−0.142 + 0.989i)5-s + (−0.142 − 0.989i)6-s + (0.415 − 0.909i)7-s + (−0.959 + 0.281i)8-s + (0.415 − 0.909i)9-s + (0.841 + 0.540i)10-s + (−0.142 + 0.989i)11-s + (−0.959 − 0.281i)12-s + (−0.959 − 0.281i)13-s + (−0.654 − 0.755i)14-s + (0.415 + 0.909i)15-s + (−0.142 + 0.989i)16-s + (−0.654 + 0.755i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.0327 - 0.999i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.0327 - 0.999i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(67\)
Sign: $0.0327 - 0.999i$
Motivic weight: \(0\)
Character: $\chi_{67} (15, \cdot )$
Sato-Tate group: $\mu(11)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 67,\ (0:\ ),\ 0.0327 - 0.999i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(0.9265065455 - 0.8966138546i\)
\(L(\frac12,\chi)\) \(\approx\) \(0.9265065455 - 0.8966138546i\)
\(L(\chi,1)\) \(\approx\) \(1.154736319 - 0.7462881553i\)
\(L(1,\chi)\) \(\approx\) \(1.154736319 - 0.7462881553i\)

Euler product

   \(L(\chi,s) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)
   \(L(s,\chi) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−32.102381244409732317793170307847, −31.59984257367058543665679335648, −30.75529995601635374577301219710, −28.89667269216094828414607193495, −27.376275708410634624589269185643, −26.86492581303449476094113428149, −25.381122967115446329319715539102, −24.6387739418190318962279026485, −23.90665346618277833543390387509, −21.97707963395967888873462539266, −21.472547188816554394687581287605, −20.17568684801332705161259907859, −18.77581145200831569746419705534, −17.21502614895044618781756132805, −16.02508946916506972185501977962, −15.33513354538160978083968875875, −14.07144593464437148946256049149, −13.06808317331795903045278886334, −11.64634918382107773496452502658, −9.24430368759659639710539780830, −8.73639744088836855505346643354, −7.47242974824482681993855829939, −5.38956917427975994286106851963, −4.55681601998545573653826544306, −2.819507937899483640549572727486, 1.832553867279940219377277116580, 3.137423313065636146069272443230, 4.46774474322589743644578296728, 6.710415251984228378236218628665, 7.92059243258007878627687443123, 9.74025345137897038881262261300, 10.6810520957010502522349457097, 12.17552431631356793530272552833, 13.291621573769643405364660479307, 14.521910108235432687765210857060, 14.95602727957930458274557705378, 17.57914145715764668835498022181, 18.483319807064016151298656943475, 19.70393427811945550061965011446, 20.285307328809664858236509816973, 21.60001806953724970996465804410, 22.91296542873672696353643833393, 23.71460737083157920966324891594, 25.09576856513800319191442886474, 26.60273820832482033987299212322, 27.10413891049475410003637321952, 28.926947807810552954240743542772, 29.86205592473782016020743583446, 30.69249110826127471059961381473, 31.20620593665573088837853852763

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