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Properties

Label 1-517-517.153-r1-0-0
Degree $1$
Conductor $517$
Sign $-0.992 - 0.125i$
Analytic cond. $55.5593$
Root an. cond. $55.5593$
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  + (−0.854 + 0.519i)2-s + (−0.576 + 0.816i)3-s + (0.460 − 0.887i)4-s + (0.203 + 0.979i)5-s + (0.0682 − 0.997i)6-s + (−0.962 + 0.269i)7-s + (0.0682 + 0.997i)8-s + (−0.334 − 0.942i)9-s + (−0.682 − 0.730i)10-s + (0.460 + 0.887i)12-s + (0.775 − 0.631i)13-s + (0.682 − 0.730i)14-s + (−0.917 − 0.398i)15-s + (−0.576 − 0.816i)16-s + (0.990 − 0.136i)17-s + (0.775 + 0.631i)18-s + ⋯
L(s)  = 1  + (−0.854 + 0.519i)2-s + (−0.576 + 0.816i)3-s + (0.460 − 0.887i)4-s + (0.203 + 0.979i)5-s + (0.0682 − 0.997i)6-s + (−0.962 + 0.269i)7-s + (0.0682 + 0.997i)8-s + (−0.334 − 0.942i)9-s + (−0.682 − 0.730i)10-s + (0.460 + 0.887i)12-s + (0.775 − 0.631i)13-s + (0.682 − 0.730i)14-s + (−0.917 − 0.398i)15-s + (−0.576 − 0.816i)16-s + (0.990 − 0.136i)17-s + (0.775 + 0.631i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 517 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(517\)    =    \(11 \cdot 47\)
Sign: $-0.992 - 0.125i$
Analytic conductor: \(55.5593\)
Root analytic conductor: \(55.5593\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{517} (153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 517,\ (1:\ ),\ -0.992 - 0.125i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.05209226513 + 0.8262715326i\)
\(L(\frac12)\) \(\approx\) \(-0.05209226513 + 0.8262715326i\)
\(L(1)\) \(\approx\) \(0.4533095185 + 0.4063896640i\)
\(L(1)\) \(\approx\) \(0.4533095185 + 0.4063896640i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
47 \( 1 \)
good2 \( 1 + (-0.854 + 0.519i)T \)
3 \( 1 + (-0.576 + 0.816i)T \)
5 \( 1 + (0.203 + 0.979i)T \)
7 \( 1 + (-0.962 + 0.269i)T \)
13 \( 1 + (0.775 - 0.631i)T \)
17 \( 1 + (0.990 - 0.136i)T \)
19 \( 1 + (-0.203 + 0.979i)T \)
23 \( 1 + (0.854 + 0.519i)T \)
29 \( 1 + (0.775 + 0.631i)T \)
31 \( 1 + (-0.576 - 0.816i)T \)
37 \( 1 + (0.682 + 0.730i)T \)
41 \( 1 + (0.0682 - 0.997i)T \)
43 \( 1 + (-0.460 + 0.887i)T \)
53 \( 1 + (-0.0682 + 0.997i)T \)
59 \( 1 + (0.460 + 0.887i)T \)
61 \( 1 + (-0.682 + 0.730i)T \)
67 \( 1 + (0.962 + 0.269i)T \)
71 \( 1 + (0.854 + 0.519i)T \)
73 \( 1 + (0.334 - 0.942i)T \)
79 \( 1 + (0.917 + 0.398i)T \)
83 \( 1 + (0.990 + 0.136i)T \)
89 \( 1 + (0.203 + 0.979i)T \)
97 \( 1 + (-0.576 + 0.816i)T \)
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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

−23.09899599670209426296480265464, −21.85339758101755337672731047755, −21.16110976869903749780797406133, −20.10062208096684401030818684185, −19.44182333995776735023971222325, −18.75608126494334431726218607486, −17.86168278844101011326559603465, −16.9505769973862984801760314878, −16.51378481686420493276392491006, −15.77014135883440934878647516552, −13.87921718977959948207305837191, −12.99068988390648196512754796631, −12.581190212964114256947432747447, −11.630827086341151086522830215731, −10.7564085641165263780704405363, −9.71678396611583315070934472594, −8.86160155774714182837589309773, −8.01723271769404482912632229688, −6.89127113147264441239125869732, −6.22751262255486736469864544577, −4.88370850332343908511436644049, −3.53861623956415090346210239771, −2.25134261167779108834780580407, −1.1007250624213930644878397282, −0.41737072668119186984979867929, 1.02745154610937077648822628451, 2.80360904802232693948153551424, 3.661933877839189179312241592773, 5.407307658290757995658542416208, 6.01076559871824048894454306386, 6.754025368311241554358765906810, 7.89275791859572030602082246803, 9.11748984022156920382639070079, 9.88729840480146211564945168448, 10.47786281111190255598669972395, 11.23663225480465513127687356140, 12.32595820988946889162430631, 13.74949124102005022719052201337, 14.86859622311451180450944321008, 15.32159487050471646566476916598, 16.28075261847642891656565020293, 16.8248699325818719759209265688, 17.9060422376045887641331620267, 18.53249503340697400887250745835, 19.27712354455548216354916929660, 20.401841006615042709145810172428, 21.270968444755600276178825897798, 22.28018723931606082103821218317, 23.09947276745096650749920142568, 23.40484272849609255689255259172

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