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Properties

Label 1-153-153.76-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: $\chi_{153} (76, \cdot )$
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

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

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