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Properties

Label 1-167-167.31-r0-0-0
Degree $1$
Conductor $167$
Sign $0.00966 - 0.999i$
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  + (−0.387 − 0.922i)2-s + (0.974 − 0.225i)3-s + (−0.700 + 0.713i)4-s + (−0.965 − 0.261i)5-s + (−0.584 − 0.811i)6-s + (0.988 − 0.150i)7-s + (0.929 + 0.369i)8-s + (0.898 − 0.438i)9-s + (0.132 + 0.991i)10-s + (0.421 + 0.906i)11-s + (−0.521 + 0.853i)12-s + (0.553 − 0.832i)13-s + (−0.521 − 0.853i)14-s + (−0.999 − 0.0378i)15-s + (−0.0189 − 0.999i)16-s + (−0.0944 − 0.995i)17-s + ⋯
L(s)  = 1  + (−0.387 − 0.922i)2-s + (0.974 − 0.225i)3-s + (−0.700 + 0.713i)4-s + (−0.965 − 0.261i)5-s + (−0.584 − 0.811i)6-s + (0.988 − 0.150i)7-s + (0.929 + 0.369i)8-s + (0.898 − 0.438i)9-s + (0.132 + 0.991i)10-s + (0.421 + 0.906i)11-s + (−0.521 + 0.853i)12-s + (0.553 − 0.832i)13-s + (−0.521 − 0.853i)14-s + (−0.999 − 0.0378i)15-s + (−0.0189 − 0.999i)16-s + (−0.0944 − 0.995i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00966 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 167 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00966 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(167\)
Sign: $0.00966 - 0.999i$
Analytic conductor: \(0.775544\)
Root analytic conductor: \(0.775544\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{167} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 167,\ (0:\ ),\ 0.00966 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8262489923 - 0.8183018254i\)
\(L(\frac12)\) \(\approx\) \(0.8262489923 - 0.8183018254i\)
\(L(1)\) \(\approx\) \(0.9326290363 - 0.5602109104i\)
\(L(1)\) \(\approx\) \(0.9326290363 - 0.5602109104i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad167 \( 1 \)
good2 \( 1 + (-0.387 - 0.922i)T \)
3 \( 1 + (0.974 - 0.225i)T \)
5 \( 1 + (-0.965 - 0.261i)T \)
7 \( 1 + (0.988 - 0.150i)T \)
11 \( 1 + (0.421 + 0.906i)T \)
13 \( 1 + (0.553 - 0.832i)T \)
17 \( 1 + (-0.0944 - 0.995i)T \)
19 \( 1 + (-0.942 + 0.334i)T \)
23 \( 1 + (-0.455 - 0.890i)T \)
29 \( 1 + (-0.455 + 0.890i)T \)
31 \( 1 + (0.280 - 0.959i)T \)
37 \( 1 + (0.898 + 0.438i)T \)
41 \( 1 + (-0.843 - 0.537i)T \)
43 \( 1 + (0.489 + 0.872i)T \)
47 \( 1 + (-0.914 - 0.404i)T \)
53 \( 1 + (0.822 + 0.569i)T \)
59 \( 1 + (-0.0944 + 0.995i)T \)
61 \( 1 + (0.206 - 0.978i)T \)
67 \( 1 + (-0.965 + 0.261i)T \)
71 \( 1 + (-0.800 + 0.599i)T \)
73 \( 1 + (-0.0189 + 0.999i)T \)
79 \( 1 + (0.0567 + 0.998i)T \)
83 \( 1 + (-0.387 + 0.922i)T \)
89 \( 1 + (-0.999 + 0.0378i)T \)
97 \( 1 + (0.280 + 0.959i)T \)
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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

−27.49745453330885553114364579087, −26.79767258283714885939575378012, −26.0909380953510345819517480241, −25.05116539856732955677143561694, −23.97360900694657852941257945551, −23.65339661683246568330635348136, −21.99986424996247815159193961156, −21.12173646120043754552829473714, −19.58613291722501713011925903418, −19.19337483061047692708512363727, −18.15576075628390010380832685163, −16.822748085881750198588224881319, −15.81079908190968594573660637420, −14.98444217193462647163491789076, −14.3164190196872420078178908432, −13.31333653876877262249159815803, −11.509924027169219154777062632661, −10.499252083423104956363441075111, −8.91615184968130432357099745140, −8.3916277342935575629206641048, −7.520855987312168775454883761900, −6.24255926883255917562458547491, −4.53547860454527230538775535939, −3.72628390087761910349052489020, −1.67020402809802359016532824136, 1.211119716793408136581961121359, 2.52112856057565673544243651333, 3.89616763886244063137350533956, 4.64692640881523337551792245550, 7.243530312930178900069845501510, 8.10447741717698185983365418110, 8.78469656545651438025516218095, 10.074738440571055656382956227557, 11.2572939947065745089797036097, 12.264723275620442042162519327284, 13.11959471590445856554284320844, 14.39418693312561689395000824347, 15.24457661138986960304711920386, 16.68006978188150101873542109102, 18.01947583978180140699737199643, 18.64559817955093124292769954290, 19.90561859609242472311253932975, 20.35483237197875452901740912061, 21.006015164717382955242330081643, 22.48897071371303208690661737124, 23.46273971609909971023414556600, 24.63540088214801049706883044348, 25.62592644451233302230203700562, 26.68721290647495459461748896563, 27.6162757526349321823884311975

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