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Properties

Label 1-671-671.172-r0-0-0
Degree $1$
Conductor $671$
Sign $0.0543 - 0.998i$
Analytic cond. $3.11611$
Root an. cond. $3.11611$
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  + (−0.951 + 0.309i)2-s + (0.809 + 0.587i)3-s + (0.809 − 0.587i)4-s + (−0.309 + 0.951i)5-s + (−0.951 − 0.309i)6-s + (−0.587 − 0.809i)7-s + (−0.587 + 0.809i)8-s + (0.309 + 0.951i)9-s − i·10-s + 12-s + (−0.309 − 0.951i)13-s + (0.809 + 0.587i)14-s + (−0.809 + 0.587i)15-s + (0.309 − 0.951i)16-s + (−0.951 − 0.309i)17-s + (−0.587 − 0.809i)18-s + ⋯
L(s)  = 1  + (−0.951 + 0.309i)2-s + (0.809 + 0.587i)3-s + (0.809 − 0.587i)4-s + (−0.309 + 0.951i)5-s + (−0.951 − 0.309i)6-s + (−0.587 − 0.809i)7-s + (−0.587 + 0.809i)8-s + (0.309 + 0.951i)9-s − i·10-s + 12-s + (−0.309 − 0.951i)13-s + (0.809 + 0.587i)14-s + (−0.809 + 0.587i)15-s + (0.309 − 0.951i)16-s + (−0.951 − 0.309i)17-s + (−0.587 − 0.809i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0543 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0543 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(671\)    =    \(11 \cdot 61\)
Sign: $0.0543 - 0.998i$
Analytic conductor: \(3.11611\)
Root analytic conductor: \(3.11611\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{671} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 671,\ (0:\ ),\ 0.0543 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2440174949 - 0.2311061084i\)
\(L(\frac12)\) \(\approx\) \(0.2440174949 - 0.2311061084i\)
\(L(1)\) \(\approx\) \(0.6127493794 + 0.1377445722i\)
\(L(1)\) \(\approx\) \(0.6127493794 + 0.1377445722i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
61 \( 1 \)
good2 \( 1 + (-0.951 + 0.309i)T \)
3 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 + (-0.309 + 0.951i)T \)
7 \( 1 + (-0.587 - 0.809i)T \)
13 \( 1 + (-0.309 - 0.951i)T \)
17 \( 1 + (-0.951 - 0.309i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 - iT \)
29 \( 1 + (0.587 + 0.809i)T \)
31 \( 1 + (-0.951 + 0.309i)T \)
37 \( 1 + (-0.587 - 0.809i)T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 - iT \)
47 \( 1 + (-0.809 - 0.587i)T \)
53 \( 1 + (0.951 - 0.309i)T \)
59 \( 1 + (-0.587 - 0.809i)T \)
67 \( 1 + iT \)
71 \( 1 + (0.951 + 0.309i)T \)
73 \( 1 + (0.809 - 0.587i)T \)
79 \( 1 + (-0.951 + 0.309i)T \)
83 \( 1 + (-0.309 + 0.951i)T \)
89 \( 1 + iT \)
97 \( 1 + (-0.309 - 0.951i)T \)
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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

−23.156512114787726953518420267674, −21.60498400292263004247776690363, −21.255471912274551904263126085, −20.16937558369300040613478216338, −19.57693871017406367559834146548, −19.07424411976951998536090095774, −18.27921269819694756011459955908, −17.2990489142702632642945782487, −16.50135593063756350200252396975, −15.58027821752215767674479569750, −14.991700553545849232499960489693, −13.51682381224423677826253153193, −12.79658123200649497082694591552, −12.09690661484733532082666462625, −11.4188681510032302483095873637, −9.86341356170054486486643332159, −9.24087917953014415981979410934, −8.59729516198067320707493139441, −7.93241942837621181838646576178, −6.84141671397272044204428681929, −6.044334064027928198876668105628, −4.34959103204252824323264786070, −3.33336316285779046730114011050, −2.17834810368453558852247478940, −1.51034278827790874956810744052, 0.186931895323585987977284077940, 2.148394104002780534538434500689, 2.950879659356451103478355095939, 3.907661715382522587495797284640, 5.21629784598008475872273785547, 6.77414600389956123267871804939, 7.06310552119956208526867463674, 8.17279280589621280244361907343, 8.90907128211480754729518178664, 9.96766476388566952648813707700, 10.593681960715993177483012702065, 11.003472751612693599109568468100, 12.57346607202889358425662332247, 13.73353897474280511442864317614, 14.548288645502368911376939799072, 15.27937603192651439979627785390, 15.876180522122432325953928420501, 16.74222898603118213578086416669, 17.70633294118101143882464969844, 18.53990930363624153891339751475, 19.48082826779720249665895553322, 19.890417889149934157064124744857, 20.53535640654057069406740044016, 21.758563079989740162330307329792, 22.53711658976587504703658764332

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