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Properties

Label 1-153-153.151-r0-0-0
Degree $1$
Conductor $153$
Sign $0.715 - 0.699i$
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  + (0.866 − 0.5i)2-s + (0.5 − 0.866i)4-s + (0.258 + 0.965i)5-s + (0.258 − 0.965i)7-s − i·8-s + (0.707 + 0.707i)10-s + (−0.258 + 0.965i)11-s + (0.5 − 0.866i)13-s + (−0.258 − 0.965i)14-s + (−0.5 − 0.866i)16-s + i·19-s + (0.965 + 0.258i)20-s + (0.258 + 0.965i)22-s + (0.965 − 0.258i)23-s + (−0.866 + 0.5i)25-s − i·26-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.5 − 0.866i)4-s + (0.258 + 0.965i)5-s + (0.258 − 0.965i)7-s − i·8-s + (0.707 + 0.707i)10-s + (−0.258 + 0.965i)11-s + (0.5 − 0.866i)13-s + (−0.258 − 0.965i)14-s + (−0.5 − 0.866i)16-s + i·19-s + (0.965 + 0.258i)20-s + (0.258 + 0.965i)22-s + (0.965 − 0.258i)23-s + (−0.866 + 0.5i)25-s − i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.715 - 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.715 - 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $0.715 - 0.699i$
Analytic conductor: \(0.710529\)
Root analytic conductor: \(0.710529\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{153} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 153,\ (0:\ ),\ 0.715 - 0.699i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.727766795 - 0.7042467891i\)
\(L(\frac12)\) \(\approx\) \(1.727766795 - 0.7042467891i\)
\(L(1)\) \(\approx\) \(1.627486178 - 0.4678887172i\)
\(L(1)\) \(\approx\) \(1.627486178 - 0.4678887172i\)

Euler product

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

−28.378405654419700257011427460982, −27.03327652541167790043963555386, −25.84004492271639825926810104352, −24.98297645185382124225286732172, −24.15690402765880246474699382219, −23.56299336703732839787269000124, −22.03836101753429390041530614682, −21.40132070433677678042444301835, −20.688367454747963473986933226991, −19.29399009406416257397436248941, −17.97606080760412829390323244158, −16.80286658146455475208892946184, −16.0453710329869497221898046966, −15.10734743575244092294746976226, −13.82160756956026662656252748764, −13.08650471521553448512838796284, −11.99135974988506193411406972378, −11.117639206655412575699150912374, −8.980079975431535405852985372587, −8.515794145192471896995243645185, −6.9005325443491644942686590745, −5.60838567016903560940406857090, −4.96898108788390630230084271383, −3.46841208677565781768285222504, −1.94844898382228068528322992609, 1.62798305888753376657100596965, 3.043684960080100131522382451022, 4.11875406182920444238950352164, 5.48163066321243348521787622002, 6.71512837334890440706616797738, 7.68620654466162933909777027315, 9.86091476741630087765077018223, 10.51848482954489344893153412115, 11.41453843111104815049309413317, 12.81922013473652674030014795043, 13.64233856143213575143627410812, 14.70821657238014088706786531005, 15.33656640494776752961074377177, 16.907517752980387681405237597283, 18.140348572882738301331812095352, 19.05188096257783965666849188729, 20.42218623344554153892060561828, 20.7829571911741454960291252658, 22.227661604052912792545413648187, 22.90797071151450604725625544185, 23.54513900262730254044667339369, 24.9222558629244465530731349591, 25.80752228898530342624256036943, 27.00309566355073904309627610322, 28.007425796385457871809345222263

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