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Properties

Label 1-6075-6075.3323-r0-0-0
Degree $1$
Conductor $6075$
Sign $0.865 - 0.500i$
Analytic cond. $28.2121$
Root an. cond. $28.2121$
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  + (0.768 − 0.639i)2-s + (0.181 − 0.983i)4-s + (−0.995 − 0.0968i)7-s + (−0.489 − 0.871i)8-s + (0.730 − 0.683i)11-s + (0.503 + 0.864i)13-s + (−0.826 + 0.562i)14-s + (−0.934 − 0.356i)16-s + (−0.606 + 0.795i)17-s + (0.651 − 0.758i)19-s + (0.123 − 0.992i)22-s + (0.737 + 0.674i)23-s + (0.939 + 0.342i)26-s + (−0.275 + 0.961i)28-s + (−0.526 − 0.850i)29-s + ⋯
L(s)  = 1  + (0.768 − 0.639i)2-s + (0.181 − 0.983i)4-s + (−0.995 − 0.0968i)7-s + (−0.489 − 0.871i)8-s + (0.730 − 0.683i)11-s + (0.503 + 0.864i)13-s + (−0.826 + 0.562i)14-s + (−0.934 − 0.356i)16-s + (−0.606 + 0.795i)17-s + (0.651 − 0.758i)19-s + (0.123 − 0.992i)22-s + (0.737 + 0.674i)23-s + (0.939 + 0.342i)26-s + (−0.275 + 0.961i)28-s + (−0.526 − 0.850i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6075 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.865 - 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6075 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.865 - 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6075\)    =    \(3^{5} \cdot 5^{2}\)
Sign: $0.865 - 0.500i$
Analytic conductor: \(28.2121\)
Root analytic conductor: \(28.2121\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6075} (3323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6075,\ (0:\ ),\ 0.865 - 0.500i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.305694493 - 0.6190871164i\)
\(L(\frac12)\) \(\approx\) \(2.305694493 - 0.6190871164i\)
\(L(1)\) \(\approx\) \(1.371545265 - 0.5395752551i\)
\(L(1)\) \(\approx\) \(1.371545265 - 0.5395752551i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (0.768 - 0.639i)T \)
7 \( 1 + (-0.995 - 0.0968i)T \)
11 \( 1 + (0.730 - 0.683i)T \)
13 \( 1 + (0.503 + 0.864i)T \)
17 \( 1 + (-0.606 + 0.795i)T \)
19 \( 1 + (0.651 - 0.758i)T \)
23 \( 1 + (0.737 + 0.674i)T \)
29 \( 1 + (-0.526 - 0.850i)T \)
31 \( 1 + (0.367 + 0.930i)T \)
37 \( 1 + (-0.964 - 0.264i)T \)
41 \( 1 + (0.345 + 0.938i)T \)
43 \( 1 + (-0.753 + 0.657i)T \)
47 \( 1 + (0.930 + 0.367i)T \)
53 \( 1 + (-0.970 + 0.241i)T \)
59 \( 1 + (-0.226 + 0.973i)T \)
61 \( 1 + (0.724 + 0.689i)T \)
67 \( 1 + (0.850 + 0.526i)T \)
71 \( 1 + (0.860 - 0.510i)T \)
73 \( 1 + (0.815 - 0.578i)T \)
79 \( 1 + (-0.143 + 0.989i)T \)
83 \( 1 + (-0.555 - 0.831i)T \)
89 \( 1 + (0.995 - 0.0929i)T \)
97 \( 1 + (0.966 + 0.256i)T \)
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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

−17.43445522000817499541780399636, −17.05139700736771775859525670915, −16.28796958106740958199816049176, −15.63148300836056189621979641846, −15.3305957320597236877087119376, −14.41533078594911556778452441534, −13.89232026492454781980665611035, −13.14301004072692845043704579325, −12.61183307131826310187102383079, −12.11109068081207059164054633952, −11.32134981998893918873158690158, −10.52380451162994484344315867709, −9.62084483500341192627786324286, −9.05429694209322268339348060812, −8.30926765548944302740938286791, −7.45893625910796970485920245856, −6.79867010742372461411348387322, −6.41384198195042289666121302435, −5.45394650237819212154732460502, −5.02472067622296763078621224319, −3.92807356346699128119110923530, −3.52303617739991937949631191640, −2.74851011037358566270633589150, −1.89717251940242466300121385578, −0.54166361275319211335128077900, 0.86659824213270472612090911565, 1.54193837169185538659130522375, 2.5389065261369580538016006794, 3.3052728340759438176599462017, 3.79647141674593043258754860817, 4.50237160017890915245100104102, 5.393175975316299957108210958970, 6.21559828473037088592222891254, 6.56577058264530898064105006004, 7.27291489007158804719405505793, 8.5636243450834270609945707739, 9.22771882826830902271214832195, 9.612785864296089889247986726, 10.57404169987356725673583761796, 11.16925518270024291405089677122, 11.681052169833256384063241317525, 12.4014244849550626720989020096, 13.20419968485945153722736577654, 13.55365290718093905159450088302, 14.13943096148455838861067862235, 14.960529195362099258951168930639, 15.68110775047931197395693528342, 16.10124348440355188058327550718, 16.925343036391318555932177045756, 17.62923045340912619957198655231

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