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Properties

Label 1-17-17.10-r1-0-0
Degree $1$
Conductor $17$
Sign $0.988 + 0.151i$
Analytic cond. $1.82690$
Root an. cond. $1.82690$
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  + (−0.707 − 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(17\)
Sign: $0.988 + 0.151i$
Analytic conductor: \(1.82690\)
Root analytic conductor: \(1.82690\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{17} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 17,\ (1:\ ),\ 0.988 + 0.151i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.117312825 + 0.08519310972i\)
\(L(\frac12)\) \(\approx\) \(1.117312825 + 0.08519310972i\)
\(L(1)\) \(\approx\) \(0.9903324006 + 0.01074511378i\)
\(L(1)\) \(\approx\) \(0.9903324006 + 0.01074511378i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
good2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 + (0.382 + 0.923i)T \)
5 \( 1 + (0.923 - 0.382i)T \)
7 \( 1 + (0.923 + 0.382i)T \)
11 \( 1 + (-0.382 + 0.923i)T \)
13 \( 1 - iT \)
19 \( 1 + (-0.707 - 0.707i)T \)
23 \( 1 + (0.382 - 0.923i)T \)
29 \( 1 + (-0.923 + 0.382i)T \)
31 \( 1 + (-0.382 - 0.923i)T \)
37 \( 1 + (0.382 + 0.923i)T \)
41 \( 1 + (0.923 + 0.382i)T \)
43 \( 1 + (-0.707 + 0.707i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.707 - 0.707i)T \)
59 \( 1 + (0.707 - 0.707i)T \)
61 \( 1 + (-0.923 - 0.382i)T \)
67 \( 1 - T \)
71 \( 1 + (0.382 + 0.923i)T \)
73 \( 1 + (0.923 - 0.382i)T \)
79 \( 1 + (-0.382 + 0.923i)T \)
83 \( 1 + (0.707 + 0.707i)T \)
89 \( 1 + iT \)
97 \( 1 + (-0.923 + 0.382i)T \)
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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

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

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