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Properties

Label 1-916-916.439-r1-0-0
Degree $1$
Conductor $916$
Sign $0.0124 - 0.999i$
Analytic cond. $98.4378$
Root an. cond. $98.4378$
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  + (−0.904 + 0.426i)3-s + (−0.962 − 0.272i)5-s + (0.451 + 0.892i)7-s + (0.635 − 0.771i)9-s + (0.677 − 0.735i)11-s + (−0.245 − 0.969i)13-s + (0.986 − 0.164i)15-s + (0.245 + 0.969i)17-s + (−0.716 − 0.697i)19-s + (−0.789 − 0.614i)21-s + (0.993 − 0.110i)23-s + (0.851 + 0.523i)25-s + (−0.245 + 0.969i)27-s + (0.998 + 0.0550i)29-s + (−0.298 − 0.954i)31-s + ⋯
L(s)  = 1  + (−0.904 + 0.426i)3-s + (−0.962 − 0.272i)5-s + (0.451 + 0.892i)7-s + (0.635 − 0.771i)9-s + (0.677 − 0.735i)11-s + (−0.245 − 0.969i)13-s + (0.986 − 0.164i)15-s + (0.245 + 0.969i)17-s + (−0.716 − 0.697i)19-s + (−0.789 − 0.614i)21-s + (0.993 − 0.110i)23-s + (0.851 + 0.523i)25-s + (−0.245 + 0.969i)27-s + (0.998 + 0.0550i)29-s + (−0.298 − 0.954i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 916 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0124 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 916 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0124 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(916\)    =    \(2^{2} \cdot 229\)
Sign: $0.0124 - 0.999i$
Analytic conductor: \(98.4378\)
Root analytic conductor: \(98.4378\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{916} (439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 916,\ (1:\ ),\ 0.0124 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5676018970 - 0.5747266276i\)
\(L(\frac12)\) \(\approx\) \(0.5676018970 - 0.5747266276i\)
\(L(1)\) \(\approx\) \(0.7153477861 + 0.001499811812i\)
\(L(1)\) \(\approx\) \(0.7153477861 + 0.001499811812i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
229 \( 1 \)
good3 \( 1 + (-0.904 + 0.426i)T \)
5 \( 1 + (-0.962 - 0.272i)T \)
7 \( 1 + (0.451 + 0.892i)T \)
11 \( 1 + (0.677 - 0.735i)T \)
13 \( 1 + (-0.245 - 0.969i)T \)
17 \( 1 + (0.245 + 0.969i)T \)
19 \( 1 + (-0.716 - 0.697i)T \)
23 \( 1 + (0.993 - 0.110i)T \)
29 \( 1 + (0.998 + 0.0550i)T \)
31 \( 1 + (-0.298 - 0.954i)T \)
37 \( 1 + (0.137 - 0.990i)T \)
41 \( 1 + (-0.350 + 0.936i)T \)
43 \( 1 + (-0.789 - 0.614i)T \)
47 \( 1 + (0.635 - 0.771i)T \)
53 \( 1 + (-0.0825 + 0.996i)T \)
59 \( 1 + (0.137 + 0.990i)T \)
61 \( 1 + (-0.986 - 0.164i)T \)
67 \( 1 + (0.350 + 0.936i)T \)
71 \( 1 + (-0.975 + 0.218i)T \)
73 \( 1 + (-0.851 - 0.523i)T \)
79 \( 1 + (-0.998 + 0.0550i)T \)
83 \( 1 + (0.926 + 0.376i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (0.0275 + 0.999i)T \)
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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

−22.10304995427312963874700260182, −21.122892574833198283037408632155, −20.232415654280227192564065428336, −19.368785859483091567670582581831, −18.80823338689417366368074810429, −17.88201575575760118270861809583, −17.06178895054500272295170392981, −16.548994114861663741832711977, −15.67669652546970531739318840469, −14.59006384479692846261184927893, −14.0087687975285668364087944262, −12.85275365946174157167535303139, −11.97754643883129150981457754938, −11.56189300769894215382038251397, −10.70458447162293549884138038210, −9.93073577924042285546207388254, −8.64802097018524211345963849377, −7.56069622489918436922909457532, −7.028209251756691193918132375551, −6.431828018315849145866778573094, −4.77015664461502589484873072178, −4.56378396831548437186798155214, −3.36948591345271971042659350404, −1.82568744772762156018829838697, −0.91464118057900500471734012435, 0.27029987791980033753042898126, 1.23121692617695398356294970908, 2.85995822885547335613441668282, 3.89494552755931178673840820134, 4.7282372479941155781313567593, 5.573863503899245273869761653516, 6.34781069220374444547355698707, 7.466493439694798612361993128525, 8.52917770739998748236017440294, 9.02879149368286405923781067733, 10.378176738674890445112490094000, 11.06248290910802844678502275728, 11.76246407084834381846825390220, 12.4085086365721038330192466321, 13.167887085739180924906664500180, 14.8661787430411104111781447717, 15.05767449231716188638816838734, 15.90329005601761364315831819240, 16.81896006866105231009143711061, 17.33579913788369298753621456093, 18.33417148498440812454739845415, 19.11129765369709446326278304270, 19.8527510375315632510215762892, 20.8729988139663750850785166765, 21.729808605976323876759662246210

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