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Properties

Label 1-604-604.87-r0-0-0
Degree $1$
Conductor $604$
Sign $-0.271 + 0.962i$
Analytic cond. $2.80496$
Root an. cond. $2.80496$
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  + (0.309 − 0.951i)3-s + (−0.809 − 0.587i)5-s + (−0.809 − 0.587i)7-s + (−0.809 − 0.587i)9-s + (−0.309 − 0.951i)11-s + (−0.309 + 0.951i)13-s + (−0.809 + 0.587i)15-s + (−0.809 + 0.587i)17-s − 19-s + (−0.809 + 0.587i)21-s + 23-s + (0.309 + 0.951i)25-s + (−0.809 + 0.587i)27-s + (0.309 + 0.951i)29-s + (0.809 − 0.587i)31-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)3-s + (−0.809 − 0.587i)5-s + (−0.809 − 0.587i)7-s + (−0.809 − 0.587i)9-s + (−0.309 − 0.951i)11-s + (−0.309 + 0.951i)13-s + (−0.809 + 0.587i)15-s + (−0.809 + 0.587i)17-s − 19-s + (−0.809 + 0.587i)21-s + 23-s + (0.309 + 0.951i)25-s + (−0.809 + 0.587i)27-s + (0.309 + 0.951i)29-s + (0.809 − 0.587i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 604 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.271 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 604 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.271 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(604\)    =    \(2^{2} \cdot 151\)
Sign: $-0.271 + 0.962i$
Analytic conductor: \(2.80496\)
Root analytic conductor: \(2.80496\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{604} (87, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 604,\ (0:\ ),\ -0.271 + 0.962i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.07335753565 - 0.09690212361i\)
\(L(\frac12)\) \(\approx\) \(-0.07335753565 - 0.09690212361i\)
\(L(1)\) \(\approx\) \(0.5915122710 - 0.3325382152i\)
\(L(1)\) \(\approx\) \(0.5915122710 - 0.3325382152i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
151 \( 1 \)
good3 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 + (-0.809 - 0.587i)T \)
7 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (-0.309 - 0.951i)T \)
13 \( 1 + (-0.309 + 0.951i)T \)
17 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 - T \)
23 \( 1 + T \)
29 \( 1 + (0.309 + 0.951i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (0.309 - 0.951i)T \)
41 \( 1 + (-0.309 + 0.951i)T \)
43 \( 1 + (0.809 - 0.587i)T \)
47 \( 1 + (-0.309 + 0.951i)T \)
53 \( 1 + (-0.309 - 0.951i)T \)
59 \( 1 - T \)
61 \( 1 + (-0.309 - 0.951i)T \)
67 \( 1 + (-0.809 + 0.587i)T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 + (-0.809 + 0.587i)T \)
89 \( 1 + (0.809 - 0.587i)T \)
97 \( 1 + (-0.809 + 0.587i)T \)
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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

−23.180570074261654376029079221954, −22.803000895923419769487920454709, −22.16446546806689691907805194590, −21.208494364430596350447763353489, −20.28326273865705658824093019513, −19.606411559110171445723764818261, −18.92591209906894576782094232118, −17.83557508173102386886862434458, −16.89887121202795486906986732361, −15.74057784418021509460954524583, −15.34211902808460160141187205680, −14.90809956526102897808771566, −13.61262749225572133277039109379, −12.61199810017743708037021321876, −11.738691705891272395113015734587, −10.669909984554803073739635586937, −10.09903836570261439734639920115, −9.11651046078526540861578450915, −8.2464352277692579801614909162, −7.22986097674462037699156639101, −6.2179763466983465092446388267, −4.94507745843280762902052281898, −4.20657163492722669417730245098, −2.960265136437239989330024581163, −2.56274986624449398259789385209, 0.0605304974075292877305349604, 1.27952156060148269251804689517, 2.66884787827087968910516309784, 3.670958399933483363764935818983, 4.63218853663745389595540424787, 6.12873222058391875861267037555, 6.81555326899232420715990014853, 7.73311824304589372480316625731, 8.63823454967705723711574107066, 9.23925373285267606003199463470, 10.77619927874035907599990994980, 11.47997442699470564533359086481, 12.62604555868594182970105220487, 13.011788899225387482408053805480, 13.89273399651009212359098079651, 14.87109775817824640502556206765, 15.91793015528650851744676720588, 16.72402885225488987900005911073, 17.361628418646167435975164964629, 18.71725760553387321982053745566, 19.316996319403223158828592932398, 19.66914221759235916538535722654, 20.67378368143284466972273004584, 21.602206226935526636157627854753, 22.802428077840411443514920693757

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