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Properties

Label 1-935-935.2-r0-0-0
Degree $1$
Conductor $935$
Sign $0.322 - 0.946i$
Analytic cond. $4.34212$
Root an. cond. $4.34212$
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  + (−0.809 − 0.587i)2-s + (−0.891 + 0.453i)3-s + (0.309 + 0.951i)4-s + (0.987 + 0.156i)6-s + (−0.891 − 0.453i)7-s + (0.309 − 0.951i)8-s + (0.587 − 0.809i)9-s + (−0.707 − 0.707i)12-s + (−0.587 + 0.809i)13-s + (0.453 + 0.891i)14-s + (−0.809 + 0.587i)16-s + (−0.951 + 0.309i)18-s + (0.951 + 0.309i)19-s + 21-s + (0.707 − 0.707i)23-s + (0.156 + 0.987i)24-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)2-s + (−0.891 + 0.453i)3-s + (0.309 + 0.951i)4-s + (0.987 + 0.156i)6-s + (−0.891 − 0.453i)7-s + (0.309 − 0.951i)8-s + (0.587 − 0.809i)9-s + (−0.707 − 0.707i)12-s + (−0.587 + 0.809i)13-s + (0.453 + 0.891i)14-s + (−0.809 + 0.587i)16-s + (−0.951 + 0.309i)18-s + (0.951 + 0.309i)19-s + 21-s + (0.707 − 0.707i)23-s + (0.156 + 0.987i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 935 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.322 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 935 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.322 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(935\)    =    \(5 \cdot 11 \cdot 17\)
Sign: $0.322 - 0.946i$
Analytic conductor: \(4.34212\)
Root analytic conductor: \(4.34212\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{935} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 935,\ (0:\ ),\ 0.322 - 0.946i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3545132094 - 0.2538449150i\)
\(L(\frac12)\) \(\approx\) \(0.3545132094 - 0.2538449150i\)
\(L(1)\) \(\approx\) \(0.4704318546 - 0.08060880890i\)
\(L(1)\) \(\approx\) \(0.4704318546 - 0.08060880890i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.809 - 0.587i)T \)
3 \( 1 + (-0.891 + 0.453i)T \)
7 \( 1 + (-0.891 - 0.453i)T \)
13 \( 1 + (-0.587 + 0.809i)T \)
19 \( 1 + (0.951 + 0.309i)T \)
23 \( 1 + (0.707 - 0.707i)T \)
29 \( 1 + (-0.891 - 0.453i)T \)
31 \( 1 + (-0.987 + 0.156i)T \)
37 \( 1 + (-0.453 + 0.891i)T \)
41 \( 1 + (0.891 - 0.453i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.951 - 0.309i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (-0.951 + 0.309i)T \)
61 \( 1 + (-0.987 - 0.156i)T \)
67 \( 1 + iT \)
71 \( 1 + (-0.156 + 0.987i)T \)
73 \( 1 + (0.453 - 0.891i)T \)
79 \( 1 + (-0.156 - 0.987i)T \)
83 \( 1 + (0.809 - 0.587i)T \)
89 \( 1 + T \)
97 \( 1 + (0.156 + 0.987i)T \)
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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

−22.43153147395343740406898647932, −21.28970385986152354882934929001, −19.95848790821987679255007706510, −19.49623702299206757754309356584, −18.59084928337547579845088149054, −18.043448293294250643193003116889, −17.277013774776380229543170640808, −16.55277702906324971803882821719, −15.840279661045782023438644563220, −15.2086850857191220044380427981, −14.08918425297258384343326117830, −13.01231584936450921289142597139, −12.38955697610723005721327708971, −11.290638688886852401075015293, −10.68340505829151798978249262658, −9.617216478448422646140593717128, −9.14576800792388993477180681626, −7.657927299150840121152667264795, −7.342841604338250656862879382541, −6.27400660455199893938974214188, −5.60543095058597251582285999810, −4.9316071078932835601369607048, −3.191497814233579141076243391413, −2.00248836908762280141533396034, −0.79370316254717676254787906891, 0.41087669264794010636656707378, 1.642588570572574395678869255611, 3.02911607514913324359370245176, 3.86018063833234434700986746155, 4.78416814420245389007277026253, 6.06957666871922554327161201575, 6.93812819576036290311123209612, 7.57902166117093383706884771111, 9.10666484914507852840027395334, 9.52118173288789314972173306578, 10.34252457335844232279495444993, 11.04399830969900487647054698222, 11.87912644386079892423994540871, 12.552646963331827370016033154370, 13.35478334755219519502292477206, 14.61944990287652330081563474406, 15.805488937901877296490110753898, 16.38773930102316699937110963457, 16.92922488600601578801903993927, 17.65041062981033376043593167650, 18.643050021121639522462391690445, 19.138925420345300721367148609617, 20.18414538576996680253871415196, 20.79900892613154181468293785956, 21.67290046818012993476401917491

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