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Properties

Label 1-96-96.83-r0-0-0
Degree $1$
Conductor $96$
Sign $0.195 + 0.980i$
Analytic cond. $0.445822$
Root an. cond. $0.445822$
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  + (−0.707 + 0.707i)5-s + i·7-s + (−0.707 + 0.707i)11-s + (0.707 + 0.707i)13-s + 17-s + (−0.707 − 0.707i)19-s − i·23-s − i·25-s + (0.707 + 0.707i)29-s − 31-s + (−0.707 − 0.707i)35-s + (0.707 − 0.707i)37-s − i·41-s + (0.707 − 0.707i)43-s − 47-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)5-s + i·7-s + (−0.707 + 0.707i)11-s + (0.707 + 0.707i)13-s + 17-s + (−0.707 − 0.707i)19-s − i·23-s − i·25-s + (0.707 + 0.707i)29-s − 31-s + (−0.707 − 0.707i)35-s + (0.707 − 0.707i)37-s − i·41-s + (0.707 − 0.707i)43-s − 47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.195 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.195 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(96\)    =    \(2^{5} \cdot 3\)
Sign: $0.195 + 0.980i$
Analytic conductor: \(0.445822\)
Root analytic conductor: \(0.445822\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{96} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 96,\ (0:\ ),\ 0.195 + 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6426714729 + 0.5274268473i\)
\(L(\frac12)\) \(\approx\) \(0.6426714729 + 0.5274268473i\)
\(L(1)\) \(\approx\) \(0.8525730541 + 0.2994947172i\)
\(L(1)\) \(\approx\) \(0.8525730541 + 0.2994947172i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.707 + 0.707i)T \)
7 \( 1 + iT \)
11 \( 1 + (-0.707 + 0.707i)T \)
13 \( 1 + (0.707 + 0.707i)T \)
17 \( 1 + T \)
19 \( 1 + (-0.707 - 0.707i)T \)
23 \( 1 - iT \)
29 \( 1 + (0.707 + 0.707i)T \)
31 \( 1 - T \)
37 \( 1 + (0.707 - 0.707i)T \)
41 \( 1 - iT \)
43 \( 1 + (0.707 - 0.707i)T \)
47 \( 1 - T \)
53 \( 1 + (0.707 - 0.707i)T \)
59 \( 1 + (0.707 - 0.707i)T \)
61 \( 1 + (-0.707 - 0.707i)T \)
67 \( 1 + (0.707 + 0.707i)T \)
71 \( 1 - iT \)
73 \( 1 + iT \)
79 \( 1 + T \)
83 \( 1 + (0.707 + 0.707i)T \)
89 \( 1 - iT \)
97 \( 1 + T \)
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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

−29.929332287905830290314142381699, −28.8970353393342390734527141823, −27.70907410922021667170848266185, −26.97351667229223233327140045923, −25.82067667211944227020311367072, −24.5755801614227388985950109081, −23.47715517013210012034642357422, −22.99597812462610129914911800189, −21.17922114049904469976019652412, −20.47387827150365302929439743928, −19.408279701722918410145235586082, −18.27498985519842380851563043633, −16.742760306138386553198077494, −16.210290457549190713593583087107, −14.82503587334289907990089984922, −13.45423156159527061307659581349, −12.56149715322979175163647669998, −11.14240820093789998826147754001, −10.14984860429370735398907943316, −8.4033516204057797091945207204, −7.72524153587122214503207717865, −5.98658384487461789415300268260, −4.51208943600151036258101005527, −3.31287004739974545725411852732, −0.91964135472425592078675018446, 2.25917560386905823972814916752, 3.67267289167868672002322377085, 5.28624176819238456067903714156, 6.73807542597732154156730350601, 7.93719984606921498898385312655, 9.22388515702069084501684008308, 10.67597771186550492038904902359, 11.7290366783488979462158638074, 12.78486733633064061582239138140, 14.35709148895770386976833598088, 15.32900977151448111242798550995, 16.14195906492624627190903545556, 17.85344332783001920967830987826, 18.68277726739775329664086606968, 19.572542789343534406727156966707, 21.040554755029738489765631188944, 21.93064141421550971252880288659, 23.198594587935384540020268540463, 23.82519222200098903068301006762, 25.51215503724210758553748695624, 25.97608651567069719879089575986, 27.456756230133173697050909178333, 28.1295972684432551564250822420, 29.31930222912836518997018012253, 30.68313332853264481532088686650

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