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Properties

Label 1-115-115.58-r1-0-0
Degree $1$
Conductor $115$
Sign $0.521 - 0.853i$
Analytic cond. $12.3584$
Root an. cond. $12.3584$
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  + (−0.909 − 0.415i)2-s + (0.281 − 0.959i)3-s + (0.654 + 0.755i)4-s + (−0.654 + 0.755i)6-s + (0.989 + 0.142i)7-s + (−0.281 − 0.959i)8-s + (−0.841 − 0.540i)9-s + (0.415 + 0.909i)11-s + (0.909 − 0.415i)12-s + (0.989 − 0.142i)13-s + (−0.841 − 0.540i)14-s + (−0.142 + 0.989i)16-s + (0.755 + 0.654i)17-s + (0.540 + 0.841i)18-s + (0.654 + 0.755i)19-s + ⋯
L(s)  = 1  + (−0.909 − 0.415i)2-s + (0.281 − 0.959i)3-s + (0.654 + 0.755i)4-s + (−0.654 + 0.755i)6-s + (0.989 + 0.142i)7-s + (−0.281 − 0.959i)8-s + (−0.841 − 0.540i)9-s + (0.415 + 0.909i)11-s + (0.909 − 0.415i)12-s + (0.989 − 0.142i)13-s + (−0.841 − 0.540i)14-s + (−0.142 + 0.989i)16-s + (0.755 + 0.654i)17-s + (0.540 + 0.841i)18-s + (0.654 + 0.755i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.521 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.521 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.521 - 0.853i$
Analytic conductor: \(12.3584\)
Root analytic conductor: \(12.3584\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 115,\ (1:\ ),\ 0.521 - 0.853i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.308503642 - 0.7340353197i\)
\(L(\frac12)\) \(\approx\) \(1.308503642 - 0.7340353197i\)
\(L(1)\) \(\approx\) \(0.9129823589 - 0.3651233975i\)
\(L(1)\) \(\approx\) \(0.9129823589 - 0.3651233975i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.909 - 0.415i)T \)
3 \( 1 + (0.281 - 0.959i)T \)
7 \( 1 + (0.989 + 0.142i)T \)
11 \( 1 + (0.415 + 0.909i)T \)
13 \( 1 + (0.989 - 0.142i)T \)
17 \( 1 + (0.755 + 0.654i)T \)
19 \( 1 + (0.654 + 0.755i)T \)
29 \( 1 + (0.654 - 0.755i)T \)
31 \( 1 + (-0.959 + 0.281i)T \)
37 \( 1 + (0.540 - 0.841i)T \)
41 \( 1 + (0.841 - 0.540i)T \)
43 \( 1 + (0.281 - 0.959i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.989 - 0.142i)T \)
59 \( 1 + (0.142 + 0.989i)T \)
61 \( 1 + (-0.959 + 0.281i)T \)
67 \( 1 + (-0.909 - 0.415i)T \)
71 \( 1 + (0.415 - 0.909i)T \)
73 \( 1 + (0.755 - 0.654i)T \)
79 \( 1 + (0.142 + 0.989i)T \)
83 \( 1 + (-0.540 + 0.841i)T \)
89 \( 1 + (0.959 + 0.281i)T \)
97 \( 1 + (-0.540 - 0.841i)T \)
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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

−28.839646476651460867743031941955, −27.71736127690517614729166860703, −27.27668576182439255184220640318, −26.316638028642449296747166904956, −25.38797561132831109442061186960, −24.34358712462698658400476987962, −23.28872874467481620150734115614, −21.7639222830995779070218630256, −20.78486426546578894853382432999, −20.00182133335896135241346250136, −18.73288441636011433108290325523, −17.679866569647711720642096486698, −16.51538426246990254897029390889, −15.872599363372837072142102968570, −14.59407537184543382080141258834, −13.921542885963060877841351041798, −11.410869192625302898919455128242, −10.96488046873065109056897761016, −9.559053770629263562179034128633, −8.66794594305688820695362735682, −7.705871738286731157038588426877, −6.00163554386137357757936910327, −4.80754068627731123341996750919, −3.07396497358551814942708548585, −1.137528880455700559907020680577, 1.141503071207643125558888866202, 2.05979325104930363667711367221, 3.713907203042213260345343805390, 5.93778644050106999249405605491, 7.36877776757967120113554406605, 8.12726357830514510717865809554, 9.18708768892694407385639922580, 10.66120338746939709131946246148, 11.8382775939830505118642236534, 12.56557836964581384848118658449, 14.02477886999198576843118544490, 15.18960946362005780192567222422, 16.757982799301153133403624525851, 17.830263372817107846104634628133, 18.34074136966075304026002450678, 19.475571639075512463699599948653, 20.45408147392279107691905190976, 21.19987511452848899076546764389, 22.857530457963321697250958703943, 24.02977427546172714821796164350, 25.12556783499595364149218000157, 25.64353023320646907980458271832, 26.94422636013596634196047163938, 27.9913471670148185936140439730, 28.732154255175513160809606746020

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