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Properties

Label 1-65-65.54-r1-0-0
Degree $1$
Conductor $65$
Sign $-0.852 + 0.522i$
Analytic cond. $6.98522$
Root an. cond. $6.98522$
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  + (−0.866 − 0.5i)2-s + (0.5 − 0.866i)3-s + (0.5 + 0.866i)4-s + (−0.866 + 0.5i)6-s + (−0.866 + 0.5i)7-s − i·8-s + (−0.5 − 0.866i)9-s + (−0.866 − 0.5i)11-s + 12-s + 14-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + i·18-s + (−0.866 + 0.5i)19-s + i·21-s + (0.5 + 0.866i)22-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.5 − 0.866i)3-s + (0.5 + 0.866i)4-s + (−0.866 + 0.5i)6-s + (−0.866 + 0.5i)7-s − i·8-s + (−0.5 − 0.866i)9-s + (−0.866 − 0.5i)11-s + 12-s + 14-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + i·18-s + (−0.866 + 0.5i)19-s + i·21-s + (0.5 + 0.866i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.852 + 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.852 + 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $-0.852 + 0.522i$
Analytic conductor: \(6.98522\)
Root analytic conductor: \(6.98522\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (54, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 65,\ (1:\ ),\ -0.852 + 0.522i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.08414063069 - 0.2985629331i\)
\(L(\frac12)\) \(\approx\) \(-0.08414063069 - 0.2985629331i\)
\(L(1)\) \(\approx\) \(0.4865572092 - 0.2979017796i\)
\(L(1)\) \(\approx\) \(0.4865572092 - 0.2979017796i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (-0.866 - 0.5i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.866 + 0.5i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 - iT \)
37 \( 1 + (0.866 + 0.5i)T \)
41 \( 1 + (0.866 + 0.5i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 - iT \)
53 \( 1 - T \)
59 \( 1 + (0.866 - 0.5i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 + (-0.866 + 0.5i)T \)
73 \( 1 - iT \)
79 \( 1 + T \)
83 \( 1 + iT \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 + (0.866 - 0.5i)T \)
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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

−32.67640720632937835484509460619, −31.91131183086202964669431895967, −30.32182452768976242557839002079, −28.76363086942170291224527259845, −28.10659510613845784923023839866, −26.69453057009970686636520650132, −26.15260021410422000803427771621, −25.30284152073315476895861936706, −23.80505462608888449800023832503, −22.623789576116924083274207868119, −21.083623705753482518505930964043, −19.96989414012789237077921723413, −19.18695606259445601576736646050, −17.62144073536025831815952918743, −16.45188120161841816714343386577, −15.60801149214935576939238211804, −14.58088815781784174888788761439, −13.06928517531332772080401697458, −10.82130198568305735242248503563, −10.08541164955431241909591822304, −8.92935346506739840100340450944, −7.69747463459286095881500298342, −6.16883333578983129189805974174, −4.429505667195133937985700095, −2.49213408096958021484142996696, 0.184156611401834596399900030114, 2.16569881454453647364729897818, 3.31679068596262360363389539249, 6.17091020157608462141870381630, 7.50619229212311676557243761726, 8.65322784711971363242014707715, 9.74702704876656516155782263919, 11.36170302487061033843362817953, 12.60619539678434963872692937021, 13.45138731238971218664579820455, 15.36249433806253039077380195204, 16.59491313778063454193872969549, 18.103012649387091258962052730307, 18.773497379076889418885645925875, 19.72615191459407591518132809245, 20.79246794945585953477860407856, 22.10842698813936278947176916967, 23.67731609227980160586241326017, 25.01896016729236294019958905702, 25.7570967825517919550318463687, 26.710034488443140139437988697732, 28.165986569175228684029317331492, 29.27668034430622325183761339315, 29.741334031801066803181717541365, 31.27862858414303447920605723596

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