Properties

Degree 1
Conductor 17
Sign $0.988 - 0.151i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.707 + 0.707i)2-s + (0.382 − 0.923i)3-s i·4-s + (0.923 + 0.382i)5-s + (0.382 + 0.923i)6-s + (0.923 − 0.382i)7-s + (0.707 + 0.707i)8-s + (−0.707 − 0.707i)9-s + (−0.923 + 0.382i)10-s + (−0.382 − 0.923i)11-s + (−0.923 − 0.382i)12-s + i·13-s + (−0.382 + 0.923i)14-s + (0.707 − 0.707i)15-s − 16-s + ⋯
L(s,χ)  = 1  + (−0.707 + 0.707i)2-s + (0.382 − 0.923i)3-s i·4-s + (0.923 + 0.382i)5-s + (0.382 + 0.923i)6-s + (0.923 − 0.382i)7-s + (0.707 + 0.707i)8-s + (−0.707 − 0.707i)9-s + (−0.923 + 0.382i)10-s + (−0.382 − 0.923i)11-s + (−0.923 − 0.382i)12-s + i·13-s + (−0.382 + 0.923i)14-s + (0.707 − 0.707i)15-s − 16-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (0.988 - 0.151i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 17 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (0.988 - 0.151i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(17\)
\( \varepsilon \)  =  $0.988 - 0.151i$
motivic weight  =  \(0\)
character  :  $\chi_{17} (12, \cdot )$
Sato-Tate  :  $\mu(16)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 17,\ (1:\ ),\ 0.988 - 0.151i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.117312825 - 0.08519310972i$
$L(\frac12,\chi)$  $\approx$  $1.117312825 - 0.08519310972i$
$L(\chi,1)$  $\approx$  0.9903324006 + 0.01074511378i
$L(1,\chi)$  $\approx$  0.9903324006 + 0.01074511378i

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

−40.85324334857053755903505284524, −39.64749815727321169037916371478, −38.31460750405404610307143793420, −37.17373731623919123251814561241, −36.51404791064985371718647034533, −34.44066071816064924019371288843, −33.12926773536538763144517949619, −31.52206490240973849857305490686, −30.16129926751069219733768534744, −28.39414315973199275513059497665, −27.65169401472526661775545503446, −26.07813875152469597260856160604, −25.01643976790543863355011486029, −22.19645290154881894732101226664, −21.00300263712762854430034066777, −20.2477382945753211790833811886, −18.09712616600168233542405898077, −16.92921271966480724773767472939, −15.0085048360268982488406590296, −12.96775964740942140200901487855, −10.88353387145199816664488206161, −9.60263413453277681184895574948, −8.2571959790708365471446385471, −4.84126687487047368130114228786, −2.345022443715306518815975574733, 1.718797166385639376573550559960, 5.896931342704853029364600749913, 7.44459053502773447030617023696, 8.97539189862638374979549283152, 10.98450099546169606251235773558, 13.67260073410650280581216103300, 14.56649323886179787408368738285, 16.9274988631441554750290743033, 18.09402310468985430162494787931, 19.16231500935584287199574104315, 21.06080941413932444337852697917, 23.502200478141037669300619251393, 24.53028729956336813576505747040, 25.744120121022264365656794299504, 26.89077667923311334915460737158, 28.86917612631691158725714570778, 29.91797446131558595138492087883, 31.68949797161101008667759001737, 33.46390486325304197229861899395, 34.381306342632718680706886072405, 35.97140007290062704913093066664, 36.931755584855267314123984618422, 37.73511608542739817601665433305, 40.39255356386663778831989173070, 41.45869717566851006606216648739

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