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Properties

Label 1-367-367.134-r0-0-0
Degree $1$
Conductor $367$
Sign $-0.774 + 0.632i$
Analytic cond. $1.70434$
Root an. cond. $1.70434$
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  + (0.0257 − 0.999i)2-s + (0.328 − 0.944i)3-s + (−0.998 − 0.0514i)4-s + (0.423 − 0.905i)5-s + (−0.935 − 0.352i)6-s + (−0.376 − 0.926i)7-s + (−0.0771 + 0.997i)8-s + (−0.784 − 0.620i)9-s + (−0.894 − 0.447i)10-s + (0.600 − 0.799i)11-s + (−0.376 + 0.926i)12-s + (−0.784 − 0.620i)13-s + (−0.935 + 0.352i)14-s + (−0.716 − 0.697i)15-s + (0.994 + 0.102i)16-s + (0.870 + 0.492i)17-s + ⋯
L(s)  = 1  + (0.0257 − 0.999i)2-s + (0.328 − 0.944i)3-s + (−0.998 − 0.0514i)4-s + (0.423 − 0.905i)5-s + (−0.935 − 0.352i)6-s + (−0.376 − 0.926i)7-s + (−0.0771 + 0.997i)8-s + (−0.784 − 0.620i)9-s + (−0.894 − 0.447i)10-s + (0.600 − 0.799i)11-s + (−0.376 + 0.926i)12-s + (−0.784 − 0.620i)13-s + (−0.935 + 0.352i)14-s + (−0.716 − 0.697i)15-s + (0.994 + 0.102i)16-s + (0.870 + 0.492i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.774 + 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.774 + 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(367\)
Sign: $-0.774 + 0.632i$
Analytic conductor: \(1.70434\)
Root analytic conductor: \(1.70434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{367} (134, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 367,\ (0:\ ),\ -0.774 + 0.632i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4113384498 - 1.155171880i\)
\(L(\frac12)\) \(\approx\) \(-0.4113384498 - 1.155171880i\)
\(L(1)\) \(\approx\) \(0.4332995235 - 0.9843133438i\)
\(L(1)\) \(\approx\) \(0.4332995235 - 0.9843133438i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad367 \( 1 \)
good2 \( 1 + (0.0257 - 0.999i)T \)
3 \( 1 + (0.328 - 0.944i)T \)
5 \( 1 + (0.423 - 0.905i)T \)
7 \( 1 + (-0.376 - 0.926i)T \)
11 \( 1 + (0.600 - 0.799i)T \)
13 \( 1 + (-0.784 - 0.620i)T \)
17 \( 1 + (0.870 + 0.492i)T \)
19 \( 1 + (-0.179 + 0.983i)T \)
23 \( 1 + (0.994 - 0.102i)T \)
29 \( 1 + (0.978 - 0.204i)T \)
31 \( 1 + (-0.998 - 0.0514i)T \)
37 \( 1 + (0.128 + 0.991i)T \)
41 \( 1 + (0.600 + 0.799i)T \)
43 \( 1 + (-0.179 - 0.983i)T \)
47 \( 1 + (-0.558 + 0.829i)T \)
53 \( 1 + (0.423 - 0.905i)T \)
59 \( 1 + (0.952 + 0.304i)T \)
61 \( 1 + (-0.935 + 0.352i)T \)
67 \( 1 + (0.952 - 0.304i)T \)
71 \( 1 + (-0.998 + 0.0514i)T \)
73 \( 1 + (0.916 - 0.400i)T \)
79 \( 1 + (-0.716 + 0.697i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.558 + 0.829i)T \)
97 \( 1 + (0.994 - 0.102i)T \)
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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

−25.32505907733406904585382715669, −24.74694836952202003661218672132, −23.188716443990366196642517937460, −22.60762393904665562265626470527, −21.71662093497018506318090928719, −21.40840637490867241349164998401, −19.70754593082965649806645760666, −18.978416730105450530770960431075, −17.932067684907171483002387382179, −17.104721398825555356863333130654, −16.17161019589370002922407384645, −15.258475659030297096667432124170, −14.657398112043856956186826540966, −14.11291503084768006305545020668, −12.809779387822536268132364190033, −11.601125313066231999783237285716, −10.25315064043024538571791568540, −9.382125858112019603912852747893, −9.01519186064774541792019957567, −7.46243084701435262870268875894, −6.663325678080210494574628378905, −5.51947817501986802804664699582, −4.69949234508469130665268410337, −3.42032679923624969845681799543, −2.38936002581567615241235137188, 0.76717049189605581314895837678, 1.51094882519308920767810926926, 2.930028862782833868765992826074, 3.86721986552639366106137865672, 5.240149241262773099565836872108, 6.285276108317610729117999191906, 7.76387183961811203947639851047, 8.52554225879915875110409300512, 9.54966432419187604505263534354, 10.39868440200196433401026118001, 11.67962528535194921320703855502, 12.58117349547644600079155932387, 13.06937186954720914432406347763, 13.97375719183538157995502030386, 14.64124531022514212669123745378, 16.73989613011425426736241249622, 17.04801336895885372519301395460, 18.09787375887001770764237832487, 19.24693373512644410058477691409, 19.6321944847981591356846560320, 20.519586094615659650053423998825, 21.20684191873321614339584231533, 22.383504821780849170987734642193, 23.30089006982383293236170212507, 23.99954700076214623130500237580

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