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Properties

Label 1-235-235.22-r0-0-0
Degree $1$
Conductor $235$
Sign $0.581 - 0.813i$
Analytic cond. $1.09133$
Root an. cond. $1.09133$
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  + (0.979 + 0.203i)2-s + (−0.730 − 0.682i)3-s + (0.917 + 0.398i)4-s + (−0.576 − 0.816i)6-s + (−0.631 − 0.775i)7-s + (0.816 + 0.576i)8-s + (0.0682 + 0.997i)9-s + (0.334 − 0.942i)11-s + (−0.398 − 0.917i)12-s + (−0.136 − 0.990i)13-s + (−0.460 − 0.887i)14-s + (0.682 + 0.730i)16-s + (0.942 − 0.334i)17-s + (−0.136 + 0.990i)18-s + (0.962 − 0.269i)19-s + ⋯
L(s)  = 1  + (0.979 + 0.203i)2-s + (−0.730 − 0.682i)3-s + (0.917 + 0.398i)4-s + (−0.576 − 0.816i)6-s + (−0.631 − 0.775i)7-s + (0.816 + 0.576i)8-s + (0.0682 + 0.997i)9-s + (0.334 − 0.942i)11-s + (−0.398 − 0.917i)12-s + (−0.136 − 0.990i)13-s + (−0.460 − 0.887i)14-s + (0.682 + 0.730i)16-s + (0.942 − 0.334i)17-s + (−0.136 + 0.990i)18-s + (0.962 − 0.269i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.581 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.581 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(235\)    =    \(5 \cdot 47\)
Sign: $0.581 - 0.813i$
Analytic conductor: \(1.09133\)
Root analytic conductor: \(1.09133\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{235} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 235,\ (0:\ ),\ 0.581 - 0.813i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.478420760 - 0.7606537927i\)
\(L(\frac12)\) \(\approx\) \(1.478420760 - 0.7606537927i\)
\(L(1)\) \(\approx\) \(1.421531421 - 0.3397507203i\)
\(L(1)\) \(\approx\) \(1.421531421 - 0.3397507203i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
47 \( 1 \)
good2 \( 1 + (0.979 + 0.203i)T \)
3 \( 1 + (-0.730 - 0.682i)T \)
7 \( 1 + (-0.631 - 0.775i)T \)
11 \( 1 + (0.334 - 0.942i)T \)
13 \( 1 + (-0.136 - 0.990i)T \)
17 \( 1 + (0.942 - 0.334i)T \)
19 \( 1 + (0.962 - 0.269i)T \)
23 \( 1 + (-0.979 + 0.203i)T \)
29 \( 1 + (-0.990 - 0.136i)T \)
31 \( 1 + (-0.682 - 0.730i)T \)
37 \( 1 + (0.887 + 0.460i)T \)
41 \( 1 + (0.576 + 0.816i)T \)
43 \( 1 + (0.398 - 0.917i)T \)
53 \( 1 + (-0.816 + 0.576i)T \)
59 \( 1 + (0.917 - 0.398i)T \)
61 \( 1 + (0.460 + 0.887i)T \)
67 \( 1 + (-0.631 + 0.775i)T \)
71 \( 1 + (0.203 + 0.979i)T \)
73 \( 1 + (-0.997 - 0.0682i)T \)
79 \( 1 + (-0.854 + 0.519i)T \)
83 \( 1 + (0.942 + 0.334i)T \)
89 \( 1 + (-0.962 - 0.269i)T \)
97 \( 1 + (0.730 + 0.682i)T \)
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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

−26.23648286331413492465599643602, −25.4371905378986873239844929044, −24.30019556264313124190369175674, −23.38569270320083419316846523189, −22.52710653149040736015683384258, −21.98111930466398980114241682267, −21.123828038169750341769826564524, −20.20823195088759199641347389724, −19.0931194597111727605914031410, −17.95041954583452781484804227563, −16.499228843030808093946003449910, −16.08914151563269425901168901764, −14.94640482989933323715683691861, −14.27758214506979340572312745368, −12.65371135715078554034589097417, −12.14024338756100806822083675649, −11.27712686772304493338911488347, −9.98348950328135278802515916590, −9.36937181288792346486480306771, −7.31131074745577302738627248928, −6.2047128004566038161872381188, −5.43826865587414147506831451410, −4.30630070009340133516159433639, −3.35102174171281925271720623461, −1.804089458222248002407532489054, 1.04134083526934206603784894331, 2.83398403276276023898255764423, 3.93789660972742838567917585598, 5.46262518577951964872540019438, 6.03110392741705217602278306627, 7.26240292131295335015388668559, 7.8923293887813467458809273365, 9.9367777690634429856209782658, 11.05595021605876453916082770640, 11.8465661093015829041117028295, 12.925930056925339302427719069, 13.53156744154802604699345555047, 14.46037547930686259444467228471, 15.971238421297360898476144825706, 16.53469673606143533361506026890, 17.422035220771411531873652754405, 18.68739281963077008880449397052, 19.76369327985071863325213418982, 20.54799942923957916625630918454, 22.06940603579315884040880546460, 22.398847775839507900955705801476, 23.40740253676124784264940567424, 24.0666162855697075894950725046, 24.92933390640794413020895571792, 25.80409467132560962836471575965

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