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Properties

Label 1-167-167.44-r0-0-0
Degree $1$
Conductor $167$
Sign $0.834 + 0.550i$
Analytic cond. $0.775544$
Root an. cond. $0.775544$
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  + (0.988 + 0.150i)2-s + (0.553 + 0.832i)3-s + (0.954 + 0.298i)4-s + (−0.584 − 0.811i)5-s + (0.421 + 0.906i)6-s + (0.132 − 0.991i)7-s + (0.898 + 0.438i)8-s + (−0.387 + 0.922i)9-s + (−0.455 − 0.890i)10-s + (0.206 + 0.978i)11-s + (0.280 + 0.959i)12-s + (0.997 + 0.0756i)13-s + (0.280 − 0.959i)14-s + (0.351 − 0.936i)15-s + (0.822 + 0.569i)16-s + (−0.993 + 0.113i)17-s + ⋯
L(s)  = 1  + (0.988 + 0.150i)2-s + (0.553 + 0.832i)3-s + (0.954 + 0.298i)4-s + (−0.584 − 0.811i)5-s + (0.421 + 0.906i)6-s + (0.132 − 0.991i)7-s + (0.898 + 0.438i)8-s + (−0.387 + 0.922i)9-s + (−0.455 − 0.890i)10-s + (0.206 + 0.978i)11-s + (0.280 + 0.959i)12-s + (0.997 + 0.0756i)13-s + (0.280 − 0.959i)14-s + (0.351 − 0.936i)15-s + (0.822 + 0.569i)16-s + (−0.993 + 0.113i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.834 + 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.834 + 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(167\)
Sign: $0.834 + 0.550i$
Analytic conductor: \(0.775544\)
Root analytic conductor: \(0.775544\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{167} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 167,\ (0:\ ),\ 0.834 + 0.550i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.122835645 + 0.6369794575i\)
\(L(\frac12)\) \(\approx\) \(2.122835645 + 0.6369794575i\)
\(L(1)\) \(\approx\) \(1.909083739 + 0.4172605933i\)
\(L(1)\) \(\approx\) \(1.909083739 + 0.4172605933i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad167 \( 1 \)
good2 \( 1 + (0.988 + 0.150i)T \)
3 \( 1 + (0.553 + 0.832i)T \)
5 \( 1 + (-0.584 - 0.811i)T \)
7 \( 1 + (0.132 - 0.991i)T \)
11 \( 1 + (0.206 + 0.978i)T \)
13 \( 1 + (0.997 + 0.0756i)T \)
17 \( 1 + (-0.993 + 0.113i)T \)
19 \( 1 + (-0.0944 - 0.995i)T \)
23 \( 1 + (-0.843 + 0.537i)T \)
29 \( 1 + (-0.843 - 0.537i)T \)
31 \( 1 + (-0.942 - 0.334i)T \)
37 \( 1 + (-0.387 - 0.922i)T \)
41 \( 1 + (0.776 + 0.629i)T \)
43 \( 1 + (-0.800 - 0.599i)T \)
47 \( 1 + (0.726 - 0.686i)T \)
53 \( 1 + (0.862 - 0.505i)T \)
59 \( 1 + (-0.993 - 0.113i)T \)
61 \( 1 + (0.929 - 0.369i)T \)
67 \( 1 + (-0.584 + 0.811i)T \)
71 \( 1 + (-0.169 + 0.985i)T \)
73 \( 1 + (0.822 - 0.569i)T \)
79 \( 1 + (-0.243 + 0.969i)T \)
83 \( 1 + (0.988 - 0.150i)T \)
89 \( 1 + (0.351 + 0.936i)T \)
97 \( 1 + (-0.942 + 0.334i)T \)
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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

−27.636774662495222148761252436154, −26.25349840155209890466347470571, −25.40412390656833955850521183891, −24.4407580132118605668362698125, −23.78686969164947369787268474540, −22.68941429426629598039973352945, −21.90591363001879867944637056395, −20.7407552590819696010443195153, −19.75353273571556960383189097557, −18.77586584883349157373196424138, −18.25357242291476299724316721598, −16.2524976748202531292087663975, −15.290030690901836537512095338732, −14.452238299996341809167855951041, −13.63946369427244938381546566587, −12.4967870319600255822254372077, −11.62769159049540255913172770425, −10.77579515089650337474738381632, −8.82604163933667345219719993973, −7.80708859358913080901382021120, −6.50406956111557709483483408603, −5.84964963317389605488302377066, −3.86385726842113227087210902130, −2.99160711385794210106450982850, −1.82303869050358822976199357757, 1.97546535573826174399006591594, 3.87058522683129765582512981518, 4.15740290968876387004751336591, 5.3209254813315657922649875541, 7.04190446499752033021600491112, 8.05488363248487502849725696349, 9.28865392746276694532423706502, 10.73566486627808270742765747351, 11.54527319869365724958331054362, 13.04330272488668712571626856966, 13.65124480774237205161890831969, 14.91203081004514992811259258965, 15.66954832099569207136197061744, 16.463605265626700405882554034519, 17.460745027328220794048373305698, 19.684525862619572102691445714536, 20.16650850611056281004591807113, 20.78905111333021186611273302100, 21.89341891997775202347178010669, 22.95647358724524741737646326157, 23.72579643142990892240494818074, 24.70213908053748722803776215131, 25.82837755024872884885602459722, 26.499330809107921557112053736238, 27.87682341682505155087185009594

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