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Properties

Label 1-153-153.11-r0-0-0
Degree $1$
Conductor $153$
Sign $-0.225 + 0.974i$
Analytic cond. $0.710529$
Root an. cond. $0.710529$
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  + (−0.258 + 0.965i)2-s + (−0.866 − 0.5i)4-s + (0.991 + 0.130i)5-s + (−0.991 + 0.130i)7-s + (0.707 − 0.707i)8-s + (−0.382 + 0.923i)10-s + (0.130 + 0.991i)11-s + (0.866 + 0.5i)13-s + (0.130 − 0.991i)14-s + (0.5 + 0.866i)16-s + (0.707 + 0.707i)19-s + (−0.793 − 0.608i)20-s + (−0.991 − 0.130i)22-s + (−0.793 + 0.608i)23-s + (0.965 + 0.258i)25-s + (−0.707 + 0.707i)26-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)2-s + (−0.866 − 0.5i)4-s + (0.991 + 0.130i)5-s + (−0.991 + 0.130i)7-s + (0.707 − 0.707i)8-s + (−0.382 + 0.923i)10-s + (0.130 + 0.991i)11-s + (0.866 + 0.5i)13-s + (0.130 − 0.991i)14-s + (0.5 + 0.866i)16-s + (0.707 + 0.707i)19-s + (−0.793 − 0.608i)20-s + (−0.991 − 0.130i)22-s + (−0.793 + 0.608i)23-s + (0.965 + 0.258i)25-s + (−0.707 + 0.707i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.225 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.225 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $-0.225 + 0.974i$
Analytic conductor: \(0.710529\)
Root analytic conductor: \(0.710529\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{153} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 153,\ (0:\ ),\ -0.225 + 0.974i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5958209285 + 0.7491115632i\)
\(L(\frac12)\) \(\approx\) \(0.5958209285 + 0.7491115632i\)
\(L(1)\) \(\approx\) \(0.7882024517 + 0.5122390334i\)
\(L(1)\) \(\approx\) \(0.7882024517 + 0.5122390334i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.258 + 0.965i)T \)
5 \( 1 + (0.991 + 0.130i)T \)
7 \( 1 + (-0.991 + 0.130i)T \)
11 \( 1 + (0.130 + 0.991i)T \)
13 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + (0.707 + 0.707i)T \)
23 \( 1 + (-0.793 + 0.608i)T \)
29 \( 1 + (0.608 - 0.793i)T \)
31 \( 1 + (-0.130 + 0.991i)T \)
37 \( 1 + (-0.923 + 0.382i)T \)
41 \( 1 + (-0.608 - 0.793i)T \)
43 \( 1 + (-0.965 - 0.258i)T \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (-0.707 - 0.707i)T \)
59 \( 1 + (0.258 + 0.965i)T \)
61 \( 1 + (0.991 - 0.130i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (0.923 - 0.382i)T \)
73 \( 1 + (0.382 + 0.923i)T \)
79 \( 1 + (-0.130 - 0.991i)T \)
83 \( 1 + (0.258 - 0.965i)T \)
89 \( 1 - iT \)
97 \( 1 + (-0.608 + 0.793i)T \)
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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

−28.09733947860410723731138771303, −26.69806858450185504271063977858, −26.01003278912070504109494669691, −25.05356665556171370754423857199, −23.64810010660616487563277809617, −22.32768618285383787610127021836, −21.90336171401895390631279590487, −20.72082987682014836060218888537, −19.93478138967666687038109165689, −18.77938928707187813075055666909, −18.01664252523298833446973953662, −16.89541247175067179769146813931, −15.979508508076754006304881020, −14.03906720895446168626035819168, −13.4119366758877874501970132791, −12.53598499032293285478741749876, −11.14862922049538964632900747002, −10.19699349514810398651057018276, −9.29518954257771502157600500178, −8.33314428358917213160905556891, −6.50472221312969128378364101292, −5.34401732671911986118189669580, −3.65110470302701868426417481696, −2.64685029553980459053185951067, −1.004185597806513363453077754634, 1.704884424893647585782307021351, 3.677050078883857590706811080655, 5.2651005668845016917638917225, 6.28327526781382495760264818618, 7.066060650001882556315834802728, 8.61707493585974598333406531586, 9.67687883046647839296088290659, 10.23924083968333735936971633448, 12.23585288399561754610147094258, 13.465197931024586061980550049615, 14.09447012568532337314535874095, 15.418419788354334781805955861040, 16.25551006418192879269170692592, 17.30016583995475264288805198321, 18.158025680679844813230229008868, 19.03108547354805279125148246982, 20.32932817209026925527601000911, 21.6899875502486467930746652707, 22.60913684446814848527428096467, 23.36446688741389829399098454962, 24.68324095014589932098823528184, 25.60154320451868207931993501945, 25.88312133163755049757542507542, 27.085599752987646798907535482107, 28.445006336709453298172377350534

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