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Properties

Label 1-345-345.152-r1-0-0
Degree $1$
Conductor $345$
Sign $-0.247 + 0.968i$
Analytic cond. $37.0753$
Root an. cond. $37.0753$
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  + (−0.540 − 0.841i)2-s + (−0.415 + 0.909i)4-s + (−0.755 + 0.654i)7-s + (0.989 − 0.142i)8-s + (0.841 + 0.540i)11-s + (0.755 + 0.654i)13-s + (0.959 + 0.281i)14-s + (−0.654 − 0.755i)16-s + (−0.909 + 0.415i)17-s + (0.415 − 0.909i)19-s − i·22-s + (0.142 − 0.989i)26-s + (−0.281 − 0.959i)28-s + (0.415 + 0.909i)29-s + (−0.142 − 0.989i)31-s + (−0.281 + 0.959i)32-s + ⋯
L(s)  = 1  + (−0.540 − 0.841i)2-s + (−0.415 + 0.909i)4-s + (−0.755 + 0.654i)7-s + (0.989 − 0.142i)8-s + (0.841 + 0.540i)11-s + (0.755 + 0.654i)13-s + (0.959 + 0.281i)14-s + (−0.654 − 0.755i)16-s + (−0.909 + 0.415i)17-s + (0.415 − 0.909i)19-s − i·22-s + (0.142 − 0.989i)26-s + (−0.281 − 0.959i)28-s + (0.415 + 0.909i)29-s + (−0.142 − 0.989i)31-s + (−0.281 + 0.959i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.247 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.247 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(345\)    =    \(3 \cdot 5 \cdot 23\)
Sign: $-0.247 + 0.968i$
Analytic conductor: \(37.0753\)
Root analytic conductor: \(37.0753\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{345} (152, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 345,\ (1:\ ),\ -0.247 + 0.968i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3921382068 + 0.5051420012i\)
\(L(\frac12)\) \(\approx\) \(0.3921382068 + 0.5051420012i\)
\(L(1)\) \(\approx\) \(0.6933691942 - 0.05505722838i\)
\(L(1)\) \(\approx\) \(0.6933691942 - 0.05505722838i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.540 - 0.841i)T \)
7 \( 1 + (-0.755 + 0.654i)T \)
11 \( 1 + (0.841 + 0.540i)T \)
13 \( 1 + (0.755 + 0.654i)T \)
17 \( 1 + (-0.909 + 0.415i)T \)
19 \( 1 + (0.415 - 0.909i)T \)
29 \( 1 + (0.415 + 0.909i)T \)
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 \( 1 - iT \)
53 \( 1 + (-0.755 + 0.654i)T \)
59 \( 1 + (-0.654 + 0.755i)T \)
61 \( 1 + (0.142 + 0.989i)T \)
67 \( 1 + (-0.540 - 0.841i)T \)
71 \( 1 + (-0.841 + 0.540i)T \)
73 \( 1 + (-0.909 - 0.415i)T \)
79 \( 1 + (-0.654 + 0.755i)T \)
83 \( 1 + (-0.281 + 0.959i)T \)
89 \( 1 + (0.142 - 0.989i)T \)
97 \( 1 + (0.281 + 0.959i)T \)
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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

−24.78877685880224484961212372116, −23.489018419505529707563459979341, −22.87373799659994884668544672021, −22.11387989330017369127930978506, −20.60930537320835217011657097186, −19.737458007913672261274784982888, −19.05598583912888696204695731820, −17.9774118652524770879977483975, −17.24047297658216170922528238356, −16.17236914653822031171079423599, −15.82841887998998582598084224951, −14.45989299243961113138816982184, −13.76826922310699406816479117056, −12.85076902415903466910834885398, −11.330810543076762619299955527725, −10.40523019351099635690926430609, −9.48314944755137750258583077968, −8.58922097883806647416623247265, −7.55583253104889324632421265872, −6.518304227239364206216092803304, −5.87488798811633955758317078183, −4.43552434303631787137453312958, −3.32219604134544901146404320899, −1.377200882474929146470084528703, −0.242896423794674304146210842437, 1.35955897278689044480456495673, 2.49449359318241105312537500763, 3.62903084621176671711665223778, 4.64288879142506088584093222453, 6.284428730016917692816071714284, 7.18321208195235926660643316663, 8.74011345393212711330499653927, 9.11529338988321220843620453846, 10.145802348096123707762946149327, 11.29221766293685971790590760734, 11.98034045061920351607771203085, 12.96871069312916398760283439211, 13.73442069736950070331825166200, 15.14132141350871707586606295328, 16.12055594901854629672307878510, 17.03941097296215853870723328594, 17.993041549469445596388834989566, 18.760795799566645170603754391182, 19.66875804073503189678432784537, 20.23676043527631832952455952187, 21.449002745262696842412521901350, 22.10898229080896039663603029231, 22.78314996177385716511698534683, 24.05503455375019624225872604108, 25.24120207019421839634350151202

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