Properties

Degree 1
Conductor $ 2^{2} \cdot 17 $
Sign $0.226 - 0.974i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.382 − 0.923i)3-s + (−0.923 − 0.382i)5-s + (0.923 − 0.382i)7-s + (−0.707 − 0.707i)9-s + (−0.382 − 0.923i)11-s + i·13-s + (−0.707 + 0.707i)15-s + (0.707 − 0.707i)19-s i·21-s + (0.382 + 0.923i)23-s + (0.707 + 0.707i)25-s + (−0.923 + 0.382i)27-s + (0.923 + 0.382i)29-s + (−0.382 + 0.923i)31-s − 33-s + ⋯
L(s,χ)  = 1  + (0.382 − 0.923i)3-s + (−0.923 − 0.382i)5-s + (0.923 − 0.382i)7-s + (−0.707 − 0.707i)9-s + (−0.382 − 0.923i)11-s + i·13-s + (−0.707 + 0.707i)15-s + (0.707 − 0.707i)19-s i·21-s + (0.382 + 0.923i)23-s + (0.707 + 0.707i)25-s + (−0.923 + 0.382i)27-s + (0.923 + 0.382i)29-s + (−0.382 + 0.923i)31-s − 33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.226 - 0.974i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 68 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.226 - 0.974i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(68\)    =    \(2^{2} \cdot 17\)
\( \varepsilon \)  =  $0.226 - 0.974i$
motivic weight  =  \(0\)
character  :  $\chi_{68} (39, \cdot )$
Sato-Tate  :  $\mu(16)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 68,\ (0:\ ),\ 0.226 - 0.974i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.7355505736 - 0.5843222257i$
$L(\frac12,\chi)$  $\approx$  $0.7355505736 - 0.5843222257i$
$L(\chi,1)$  $\approx$  0.9443061744 - 0.4199308384i
$L(1,\chi)$  $\approx$  0.9443061744 - 0.4199308384i

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−32.05407155458239634877225146027, −31.03181185753534237315346196962, −30.49461669828625147117906654616, −28.551113698918095228051001936308, −27.53219188844512795168478951977, −26.97137937013894714436703981030, −25.73527793401785434806002669658, −24.58025096399572127350054250903, −23.07562638261696675718215502289, −22.28427618520297255186074601274, −20.810404247939170302704655399782, −20.20708289276609101304511490603, −18.769112160384508074344542876490, −17.516893117785268874948941832346, −15.924617274045452965092161242769, −15.13971080528547196703241369671, −14.31484484114913413990603565786, −12.382005586334026035158121385398, −11.11186057373821780131180580609, −10.094658899700320153740435448953, −8.467796374136064714519430362318, −7.571936324624049233450331493534, −5.31532210919947948252396777270, −4.135073604262902445775594756, −2.63852168854888205208940764878, 1.2939612276670734417592203602, 3.29824360252817815506395422952, 4.981361421091537974968057854534, 6.91410280533428067886240571121, 7.97844637478264944047698267362, 8.90622038064882185642092124723, 11.16765827970942081584079134056, 11.93361912440944548204617045071, 13.40874219607682440901978283141, 14.32716987304950328690157212886, 15.761290852140284640287931128133, 17.10969136752935463274076412824, 18.380364191045791780462837355348, 19.38866578189246723983743720323, 20.31310425919756988637207065901, 21.52811258592806533864452498909, 23.4883878800814163254673331072, 23.84731564327044535122866859586, 24.818716774701306028602341215764, 26.34974864567622398252931512839, 27.17689547841740978044770460576, 28.5703123076298013712554958251, 29.62539711742217294342152997618, 30.92246298035136698804529331195, 31.26117005606192464087187970229

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