Properties

Degree $1$
Conductor $67$
Sign $0.994 - 0.107i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.723 + 0.690i)2-s + (−0.142 − 0.989i)3-s + (0.0475 + 0.998i)4-s + (0.415 − 0.909i)5-s + (0.580 − 0.814i)6-s + (0.235 − 0.971i)7-s + (−0.654 + 0.755i)8-s + (−0.959 + 0.281i)9-s + (0.928 − 0.371i)10-s + (0.580 + 0.814i)11-s + (0.981 − 0.189i)12-s + (−0.327 + 0.945i)13-s + (0.841 − 0.540i)14-s + (−0.959 − 0.281i)15-s + (−0.995 + 0.0950i)16-s + (0.0475 − 0.998i)17-s + ⋯
L(s,χ)  = 1  + (0.723 + 0.690i)2-s + (−0.142 − 0.989i)3-s + (0.0475 + 0.998i)4-s + (0.415 − 0.909i)5-s + (0.580 − 0.814i)6-s + (0.235 − 0.971i)7-s + (−0.654 + 0.755i)8-s + (−0.959 + 0.281i)9-s + (0.928 − 0.371i)10-s + (0.580 + 0.814i)11-s + (0.981 − 0.189i)12-s + (−0.327 + 0.945i)13-s + (0.841 − 0.540i)14-s + (−0.959 − 0.281i)15-s + (−0.995 + 0.0950i)16-s + (0.0475 − 0.998i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.994 - 0.107i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.994 - 0.107i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(67\)
Sign: $0.994 - 0.107i$
Motivic weight: \(0\)
Character: $\chi_{67} (55, \cdot )$
Sato-Tate group: $\mu(33)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 67,\ (0:\ ),\ 0.994 - 0.107i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(1.285710053 - 0.06947727385i\)
\(L(\frac12,\chi)\) \(\approx\) \(1.285710053 - 0.06947727385i\)
\(L(\chi,1)\) \(\approx\) \(1.375658067 + 0.002938546210i\)
\(L(1,\chi)\) \(\approx\) \(1.375658067 + 0.002938546210i\)

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

−32.26665587825237868755734915318, −30.88039817380503833967838296152, −29.99494865889214047182567753131, −28.75793542242878592491420414392, −27.844172329187596652780065997498, −26.816076258413186182820580562125, −25.43569097543767091577451362037, −24.12253363410172873141452255439, −22.50095820619385464293743327732, −22.03269585293754624945037368432, −21.30293832842658095669985982854, −19.93855120098455713668697556236, −18.74005212351601924176457312697, −17.43071172667464972124071032639, −15.55651830152480555004284668864, −14.88003272075852225489714449811, −13.825301177860578348610189437825, −12.110245298622091246696670053261, −11.01641860941125686249951535107, −10.14073460213497345369966012665, −8.828369446408544747540899103, −6.21319983321410020918747539369, −5.33550454595489970311731070427, −3.64055104245760379597111814691, −2.50285139473869148785406473973, 1.8067559837071901945037770574, 4.14327889509968379862388899508, 5.47021131668229920789699471945, 6.89099922970307322569058551936, 7.78668819441890235290321980588, 9.33559226027715656337045454270, 11.661275253112375854393950111249, 12.54553074261932726160697327457, 13.717139261910412673472914259364, 14.3607638379334839641883848772, 16.3960137394440873254397968796, 17.06761135433588216867855147154, 18.0660900446695629985550148424, 19.95187194699357613948516445942, 20.78934380468195502400504097745, 22.36146580562618024998569969501, 23.435220218633079039174739353601, 24.22519542111154857006103410394, 25.05834338077038012901880372195, 26.051851103688336539582390744231, 27.60519661585975021004266566455, 29.189959861448969549604262307775, 29.84611332323244177720652831510, 31.06887451430871737057589835080, 31.88930111858292451180171429417

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