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Properties

Label 1-300-300.23-r1-0-0
Degree $1$
Conductor $300$
Sign $-0.481 - 0.876i$
Analytic cond. $32.2394$
Root an. cond. $32.2394$
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  − i·7-s + (0.309 − 0.951i)11-s + (−0.951 + 0.309i)13-s + (−0.587 − 0.809i)17-s + (−0.809 + 0.587i)19-s + (0.951 + 0.309i)23-s + (−0.809 − 0.587i)29-s + (0.809 − 0.587i)31-s + (0.951 − 0.309i)37-s + (−0.309 − 0.951i)41-s − i·43-s + (−0.587 + 0.809i)47-s − 49-s + (−0.587 + 0.809i)53-s + (−0.309 − 0.951i)59-s + ⋯
L(s)  = 1  − i·7-s + (0.309 − 0.951i)11-s + (−0.951 + 0.309i)13-s + (−0.587 − 0.809i)17-s + (−0.809 + 0.587i)19-s + (0.951 + 0.309i)23-s + (−0.809 − 0.587i)29-s + (0.809 − 0.587i)31-s + (0.951 − 0.309i)37-s + (−0.309 − 0.951i)41-s − i·43-s + (−0.587 + 0.809i)47-s − 49-s + (−0.587 + 0.809i)53-s + (−0.309 − 0.951i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.481 - 0.876i$
Analytic conductor: \(32.2394\)
Root analytic conductor: \(32.2394\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 300,\ (1:\ ),\ -0.481 - 0.876i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3804286954 - 0.6432697935i\)
\(L(\frac12)\) \(\approx\) \(0.3804286954 - 0.6432697935i\)
\(L(1)\) \(\approx\) \(0.8738987604 - 0.05258475350i\)
\(L(1)\) \(\approx\) \(0.8738987604 - 0.05258475350i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - iT \)
11 \( 1 + (0.309 - 0.951i)T \)
13 \( 1 + (-0.951 + 0.309i)T \)
17 \( 1 + (-0.587 - 0.809i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (0.951 + 0.309i)T \)
29 \( 1 + (-0.809 - 0.587i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (0.951 - 0.309i)T \)
41 \( 1 + (-0.309 - 0.951i)T \)
43 \( 1 - iT \)
47 \( 1 + (-0.587 + 0.809i)T \)
53 \( 1 + (-0.587 + 0.809i)T \)
59 \( 1 + (-0.309 - 0.951i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + (-0.587 - 0.809i)T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (0.951 + 0.309i)T \)
79 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 + (-0.587 - 0.809i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (-0.587 + 0.809i)T \)
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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

−25.478797368286814222370228766309, −24.51407978160071347771490098917, −23.59539929961125306726405191926, −22.82658559736245681400422111143, −21.919074309174777383667267051532, −20.88088411400117673942175841894, −19.86042571997809389786400492668, −19.46762935301302432117000457368, −17.96589187482957945179691440827, −17.243548164557545619092844639875, −16.56028788635651830688627862533, −15.07565802443299774515995072957, −14.67361312121664875827841675644, −13.25311151712690934111490519121, −12.69065910087713742856962718819, −11.385400178436273556147648435368, −10.43339515473443336907149591350, −9.60864572603323671739355948789, −8.36844455189957726638674241247, −7.22283707422844797546887532502, −6.544656235893395874121850353120, −4.89854940318468719692757395702, −4.17701197515839104022546135012, −2.724782390297508935484454206336, −1.344875859339969019664843862817, 0.22168684505397450538819075689, 2.00714439268047298438895626955, 3.04453827357525893737308348355, 4.4722767232777459868376119687, 5.59270236476052490072083142482, 6.53528646406714613431282601707, 7.79716191701928221726919387523, 8.90172793781992468834056225475, 9.58446846630195326738148754355, 11.01359091149923983261479823715, 11.7820518606926731392085741400, 12.736682582215390749416939211844, 13.82263684647259515152295468586, 14.813814296240192891519824191603, 15.61874038797082658747708953959, 16.695636828889669889426753288308, 17.49154509440242960319576225501, 18.88481794811592669374524201583, 19.04435596005091990583290939879, 20.41417241221471749853167108940, 21.41388059451956301430420330492, 22.06205777583602548187309573417, 22.94388458456309656356193271721, 24.22402573120477341823753100343, 24.73987323841552347146624101195

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