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Properties

Label 1-4235-4235.2834-r0-0-0
Degree $1$
Conductor $4235$
Sign $-0.713 - 0.700i$
Analytic cond. $19.6672$
Root an. cond. $19.6672$
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  + (−0.0285 − 0.999i)2-s + (−0.809 − 0.587i)3-s + (−0.998 + 0.0570i)4-s + (−0.564 + 0.825i)6-s + (0.0855 + 0.996i)8-s + (0.309 + 0.951i)9-s + (0.841 + 0.540i)12-s + (0.254 − 0.967i)13-s + (0.993 − 0.113i)16-s + (0.985 + 0.170i)17-s + (0.941 − 0.336i)18-s + (0.696 − 0.717i)19-s + (−0.415 − 0.909i)23-s + (0.516 − 0.856i)24-s + (−0.974 − 0.226i)26-s + (0.309 − 0.951i)27-s + ⋯
L(s)  = 1  + (−0.0285 − 0.999i)2-s + (−0.809 − 0.587i)3-s + (−0.998 + 0.0570i)4-s + (−0.564 + 0.825i)6-s + (0.0855 + 0.996i)8-s + (0.309 + 0.951i)9-s + (0.841 + 0.540i)12-s + (0.254 − 0.967i)13-s + (0.993 − 0.113i)16-s + (0.985 + 0.170i)17-s + (0.941 − 0.336i)18-s + (0.696 − 0.717i)19-s + (−0.415 − 0.909i)23-s + (0.516 − 0.856i)24-s + (−0.974 − 0.226i)26-s + (0.309 − 0.951i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.713 - 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.713 - 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-0.713 - 0.700i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (2834, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ -0.713 - 0.700i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4444423248 - 1.086920906i\)
\(L(\frac12)\) \(\approx\) \(0.4444423248 - 1.086920906i\)
\(L(1)\) \(\approx\) \(0.6063084109 - 0.5354316035i\)
\(L(1)\) \(\approx\) \(0.6063084109 - 0.5354316035i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.0285 - 0.999i)T \)
3 \( 1 + (-0.809 - 0.587i)T \)
13 \( 1 + (0.254 - 0.967i)T \)
17 \( 1 + (0.985 + 0.170i)T \)
19 \( 1 + (0.696 - 0.717i)T \)
23 \( 1 + (-0.415 - 0.909i)T \)
29 \( 1 + (0.466 + 0.884i)T \)
31 \( 1 + (-0.774 - 0.633i)T \)
37 \( 1 + (0.362 - 0.931i)T \)
41 \( 1 + (0.610 + 0.791i)T \)
43 \( 1 + (-0.654 - 0.755i)T \)
47 \( 1 + (0.941 + 0.336i)T \)
53 \( 1 + (-0.993 - 0.113i)T \)
59 \( 1 + (-0.610 + 0.791i)T \)
61 \( 1 + (-0.0285 + 0.999i)T \)
67 \( 1 + (0.959 + 0.281i)T \)
71 \( 1 + (0.897 + 0.441i)T \)
73 \( 1 + (0.870 + 0.491i)T \)
79 \( 1 + (0.921 + 0.389i)T \)
83 \( 1 + (0.736 + 0.676i)T \)
89 \( 1 + (0.142 + 0.989i)T \)
97 \( 1 + (0.516 - 0.856i)T \)
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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

−18.51098217286444669150149966494, −17.77107481682450013760840627627, −17.010601150299139099915512336850, −16.66982974073930205209646066839, −15.822540459793590074342022936210, −15.64829278279034285058584470088, −14.548257288739317876815399776321, −14.150607812082210536879145016741, −13.38178312186500005078767398998, −12.35113619412129657705799003480, −11.934739771640156610735063188825, −11.06768612364704895370845718720, −10.18212610298901194836609330371, −9.56276306356793191155176006732, −9.15407077619450224915413011868, −8.02317568911631714364997574536, −7.50716539247565623923494129187, −6.49681727561123540821366611665, −6.12063105485182907901867646131, −5.24717390601619965022768077509, −4.79840604445851131125440066899, −3.78236419959642028100872971114, −3.41214524010640935937295007294, −1.692631281348607184408994482581, −0.77053985767282910328928146213, 0.611119138822988440162000144295, 1.14570189828084521004493264265, 2.18200295835261623840484284137, 2.911501388556611860459421159762, 3.78976640774353894005624414893, 4.7055388844440875726164904858, 5.44044511866140769261068508196, 5.91009815013592190168027840556, 7.01126990634659827736788619809, 7.80205742530314022324590182381, 8.35790253555053294606476711203, 9.36258401304206492031254972417, 10.07243043238558728207767693095, 10.816131956842913103135658172974, 11.1454986090066244893374732600, 12.12213042637578652682892344113, 12.5079647251764068783187307842, 13.07183065656637285782580611886, 13.85117250841463716951345003742, 14.454764489905654827241998236357, 15.43386510237794593326066074234, 16.38642245044452877119040673544, 16.88877076380315794280736383193, 17.77260530978008780581083286266, 18.16823227978041099686432499031

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