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Properties

Label 1-417-417.59-r0-0-0
Degree $1$
Conductor $417$
Sign $-0.749 + 0.661i$
Analytic cond. $1.93653$
Root an. cond. $1.93653$
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  + (−0.576 − 0.816i)2-s + (−0.334 + 0.942i)4-s + (−0.854 − 0.519i)5-s + (−0.775 − 0.631i)7-s + (0.962 − 0.269i)8-s + (0.0682 + 0.997i)10-s + (0.917 + 0.398i)11-s + (−0.0682 − 0.997i)13-s + (−0.0682 + 0.997i)14-s + (−0.775 − 0.631i)16-s + (0.203 − 0.979i)17-s + (−0.203 + 0.979i)19-s + (0.775 − 0.631i)20-s + (−0.203 − 0.979i)22-s + (−0.775 − 0.631i)23-s + ⋯
L(s)  = 1  + (−0.576 − 0.816i)2-s + (−0.334 + 0.942i)4-s + (−0.854 − 0.519i)5-s + (−0.775 − 0.631i)7-s + (0.962 − 0.269i)8-s + (0.0682 + 0.997i)10-s + (0.917 + 0.398i)11-s + (−0.0682 − 0.997i)13-s + (−0.0682 + 0.997i)14-s + (−0.775 − 0.631i)16-s + (0.203 − 0.979i)17-s + (−0.203 + 0.979i)19-s + (0.775 − 0.631i)20-s + (−0.203 − 0.979i)22-s + (−0.775 − 0.631i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 417 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 417 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(417\)    =    \(3 \cdot 139\)
Sign: $-0.749 + 0.661i$
Analytic conductor: \(1.93653\)
Root analytic conductor: \(1.93653\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{417} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 417,\ (0:\ ),\ -0.749 + 0.661i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.08667347158 - 0.2291058222i\)
\(L(\frac12)\) \(\approx\) \(-0.08667347158 - 0.2291058222i\)
\(L(1)\) \(\approx\) \(0.4336856148 - 0.2917757342i\)
\(L(1)\) \(\approx\) \(0.4336856148 - 0.2917757342i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
139 \( 1 \)
good2 \( 1 + (-0.576 - 0.816i)T \)
5 \( 1 + (-0.854 - 0.519i)T \)
7 \( 1 + (-0.775 - 0.631i)T \)
11 \( 1 + (0.917 + 0.398i)T \)
13 \( 1 + (-0.0682 - 0.997i)T \)
17 \( 1 + (0.203 - 0.979i)T \)
19 \( 1 + (-0.203 + 0.979i)T \)
23 \( 1 + (-0.775 - 0.631i)T \)
29 \( 1 + (0.334 - 0.942i)T \)
31 \( 1 + (-0.990 - 0.136i)T \)
37 \( 1 + (0.460 + 0.887i)T \)
41 \( 1 + (-0.682 + 0.730i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.854 + 0.519i)T \)
53 \( 1 + (-0.576 - 0.816i)T \)
59 \( 1 + (-0.334 - 0.942i)T \)
61 \( 1 + (-0.682 - 0.730i)T \)
67 \( 1 + (-0.917 + 0.398i)T \)
71 \( 1 + (-0.854 - 0.519i)T \)
73 \( 1 + (-0.962 + 0.269i)T \)
79 \( 1 + (0.682 + 0.730i)T \)
83 \( 1 + (0.917 + 0.398i)T \)
89 \( 1 + (-0.460 + 0.887i)T \)
97 \( 1 - T \)
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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

−24.83439840191375328079754420786, −23.82383765562979897853474936988, −23.42206296746766330574153139798, −22.129779378687055615117681305037, −21.81147196052816704919203645197, −19.74093743342896502015113486158, −19.565422112898594165105455885524, −18.72328763151417946613256299244, −17.883933142562093328375349116286, −16.66317330967021310487607896565, −16.15928845821612260338567526852, −15.17867558592134405532928156119, −14.58545549472190774050998007157, −13.54840006809055285142180486614, −12.22543046106039845991763136199, −11.3290301214683508520612008430, −10.33196653978472047510752527218, −9.147437631892160090809148297742, −8.652417466049044748937570777560, −7.36326797074426400583708643738, −6.62915468968583013046522215728, −5.84793869998952971748601312266, −4.392081222353143684557819283494, −3.36740106646266129979535702744, −1.73460956577608210328411400757, 0.182259467548113459064845996012, 1.42392182003894063025194018760, 3.06365207887040871781046236696, 3.84359429500773541476628387278, 4.76500770733229539946737615575, 6.50403004587815370063711035453, 7.63700959398804963656791162345, 8.28403725679356446574146157347, 9.5330275216370024745850562806, 10.08025700914217393279420560966, 11.26709898568614271545749313348, 12.122134944165621847893769770154, 12.74110481476646750151918115498, 13.71128001837321488564734203193, 14.98790042269610509525446462643, 16.29300701256154747420417072486, 16.62827225690313936862102861923, 17.69204383902831418115629657475, 18.72401707368279758297638836375, 19.52617004015489419680445683977, 20.31479915778149127868933610757, 20.545253034523400755037461737, 22.11082645988445682257117655161, 22.726961393463407146414552822621, 23.402767078808147081594052475225

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