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Properties

Label 1-368-368.133-r0-0-0
Degree $1$
Conductor $368$
Sign $0.468 - 0.883i$
Analytic cond. $1.70898$
Root an. cond. $1.70898$
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  + (−0.989 + 0.142i)3-s + (0.281 − 0.959i)5-s + (0.654 − 0.755i)7-s + (0.959 − 0.281i)9-s + (0.540 + 0.841i)11-s + (−0.755 + 0.654i)13-s + (−0.142 + 0.989i)15-s + (0.415 − 0.909i)17-s + (0.909 − 0.415i)19-s + (−0.540 + 0.841i)21-s + (−0.841 − 0.540i)25-s + (−0.909 + 0.415i)27-s + (−0.909 − 0.415i)29-s + (−0.142 + 0.989i)31-s + (−0.654 − 0.755i)33-s + ⋯
L(s)  = 1  + (−0.989 + 0.142i)3-s + (0.281 − 0.959i)5-s + (0.654 − 0.755i)7-s + (0.959 − 0.281i)9-s + (0.540 + 0.841i)11-s + (−0.755 + 0.654i)13-s + (−0.142 + 0.989i)15-s + (0.415 − 0.909i)17-s + (0.909 − 0.415i)19-s + (−0.540 + 0.841i)21-s + (−0.841 − 0.540i)25-s + (−0.909 + 0.415i)27-s + (−0.909 − 0.415i)29-s + (−0.142 + 0.989i)31-s + (−0.654 − 0.755i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.468 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.468 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(368\)    =    \(2^{4} \cdot 23\)
Sign: $0.468 - 0.883i$
Analytic conductor: \(1.70898\)
Root analytic conductor: \(1.70898\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{368} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 368,\ (0:\ ),\ 0.468 - 0.883i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8988273735 - 0.5410016724i\)
\(L(\frac12)\) \(\approx\) \(0.8988273735 - 0.5410016724i\)
\(L(1)\) \(\approx\) \(0.8926813445 - 0.2188150996i\)
\(L(1)\) \(\approx\) \(0.8926813445 - 0.2188150996i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
23 \( 1 \)
good3 \( 1 + (-0.989 + 0.142i)T \)
5 \( 1 + (0.281 - 0.959i)T \)
7 \( 1 + (0.654 - 0.755i)T \)
11 \( 1 + (0.540 + 0.841i)T \)
13 \( 1 + (-0.755 + 0.654i)T \)
17 \( 1 + (0.415 - 0.909i)T \)
19 \( 1 + (0.909 - 0.415i)T \)
29 \( 1 + (-0.909 - 0.415i)T \)
31 \( 1 + (-0.142 + 0.989i)T \)
37 \( 1 + (-0.281 - 0.959i)T \)
41 \( 1 + (0.959 + 0.281i)T \)
43 \( 1 + (0.989 - 0.142i)T \)
47 \( 1 + T \)
53 \( 1 + (-0.755 - 0.654i)T \)
59 \( 1 + (0.755 - 0.654i)T \)
61 \( 1 + (0.989 + 0.142i)T \)
67 \( 1 + (0.540 - 0.841i)T \)
71 \( 1 + (-0.841 - 0.540i)T \)
73 \( 1 + (-0.415 - 0.909i)T \)
79 \( 1 + (-0.654 - 0.755i)T \)
83 \( 1 + (0.281 + 0.959i)T \)
89 \( 1 + (0.142 + 0.989i)T \)
97 \( 1 + (-0.959 - 0.281i)T \)
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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

−24.55379106750387577051557048856, −24.09132381027479459168935405957, −22.85329093518521553981752875173, −22.05438173581646417525505721659, −21.8029124414445241162577968565, −20.6086487463796472200960443524, −19.03985802735303336930750837086, −18.70417749189139352624433802035, −17.64038418659841199392829507472, −17.126480619337827353995080834996, −15.93919077324629944332463898732, −14.93852991123617177596898622187, −14.23606203827807821291033137059, −12.96825084407672064502204310431, −11.958031002340843738817131271881, −11.266342051412170316903416548893, −10.44802453719987676360615697210, −9.46251504860466075555305914865, −8.01062079978994034161487360577, −7.1310911591615532841400856770, −5.7896211245181269414467370267, −5.61520929746607857200184375077, −3.99469393605871123814365351872, −2.655643227843089252278853774, −1.35710645833104511233713587159, 0.83415663235702529930876607680, 1.902904050020926239681471091290, 4.03802658554863112809380741751, 4.7968806416260860282018633958, 5.49209235737543674739894878215, 6.95672177730467278424348843174, 7.58598546564540058146809901642, 9.26728868089461874499889212396, 9.76896915178017492848526149004, 11.01532366131480531573206942600, 11.90249210011967762280478159268, 12.535991151294085519243040035286, 13.71692813289575770173643366041, 14.60469970037227247957375743746, 15.97634033280474329188792515299, 16.58574491659170463576881552355, 17.50922107736202982084839012328, 17.815961042794647213540327690720, 19.2848371845699751488456920038, 20.385578073714175080041838967726, 20.92515914328157772379304904229, 21.94841108825165445896598382320, 22.802962680439236757691887831519, 23.68976904700297949799042865531, 24.37456335031850183939846277949

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