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Properties

Label 1-1480-1480.1069-r0-0-0
Degree $1$
Conductor $1480$
Sign $0.0357 + 0.999i$
Analytic cond. $6.87309$
Root an. cond. $6.87309$
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  + (−0.939 + 0.342i)3-s + (−0.173 + 0.984i)7-s + (0.766 − 0.642i)9-s + (0.5 + 0.866i)11-s + (0.766 + 0.642i)13-s + (−0.766 + 0.642i)17-s + (0.939 − 0.342i)19-s + (−0.173 − 0.984i)21-s + (0.5 − 0.866i)23-s + (−0.5 + 0.866i)27-s + (0.5 + 0.866i)29-s + 31-s + (−0.766 − 0.642i)33-s + (−0.939 − 0.342i)39-s + (0.766 + 0.642i)41-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)3-s + (−0.173 + 0.984i)7-s + (0.766 − 0.642i)9-s + (0.5 + 0.866i)11-s + (0.766 + 0.642i)13-s + (−0.766 + 0.642i)17-s + (0.939 − 0.342i)19-s + (−0.173 − 0.984i)21-s + (0.5 − 0.866i)23-s + (−0.5 + 0.866i)27-s + (0.5 + 0.866i)29-s + 31-s + (−0.766 − 0.642i)33-s + (−0.939 − 0.342i)39-s + (0.766 + 0.642i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0357 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0357 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1480\)    =    \(2^{3} \cdot 5 \cdot 37\)
Sign: $0.0357 + 0.999i$
Analytic conductor: \(6.87309\)
Root analytic conductor: \(6.87309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1480} (1069, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1480,\ (0:\ ),\ 0.0357 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8880793782 + 0.8569181084i\)
\(L(\frac12)\) \(\approx\) \(0.8880793782 + 0.8569181084i\)
\(L(1)\) \(\approx\) \(0.8505025359 + 0.3085256513i\)
\(L(1)\) \(\approx\) \(0.8505025359 + 0.3085256513i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
37 \( 1 \)
good3 \( 1 + (-0.939 + 0.342i)T \)
7 \( 1 + (-0.173 + 0.984i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.766 + 0.642i)T \)
17 \( 1 + (-0.766 + 0.642i)T \)
19 \( 1 + (0.939 - 0.342i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + T \)
41 \( 1 + (0.766 + 0.642i)T \)
43 \( 1 + T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (0.173 + 0.984i)T \)
59 \( 1 + (-0.173 - 0.984i)T \)
61 \( 1 + (-0.766 - 0.642i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (-0.939 + 0.342i)T \)
73 \( 1 - T \)
79 \( 1 + (0.173 - 0.984i)T \)
83 \( 1 + (0.766 - 0.642i)T \)
89 \( 1 + (0.173 + 0.984i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−20.58058425676883875085218333008, −19.50159961779925895366102078267, −19.08804422828541596067443049382, −17.94217764391580592845823963510, −17.59703628658484932536871913298, −16.76500133838522239683362463686, −16.02550209323848448590010504110, −15.56304551347154563603543174885, −14.06952006716094570067424298445, −13.59897621923881670174945215378, −12.99711324064879663307519994018, −11.88204666824923092483601471528, −11.316603952656487447741090893987, −10.65438334620093690052211453336, −9.88044451633645688614490363397, −8.87682042430944018800197340687, −7.76623499871912977598227190697, −7.19003842749159685881104592359, −6.22950617762934635328819649128, −5.69977322726020009994435399712, −4.605686975352573789086763483555, −3.808647693988698770049909194883, −2.78430022337234068257215477056, −1.23231501802894679902972221697, −0.720471420487009101981728356345, 1.074453625707294458878136482604, 2.10762888339736044757058120510, 3.29386238699993487788257074019, 4.4156129373421439467253617390, 4.90529952766842551086873591762, 6.09854753146451745139200399599, 6.44760254186540324231886833062, 7.401398995243270409860910344614, 8.79492800220352323573934483868, 9.18537030261197108599297356311, 10.14810214281213418503275230659, 10.97599819391509862031309090108, 11.70286528042906109756163776067, 12.33750864093709464279617851309, 12.98601632062445804534966416293, 14.11278167358343493383305130219, 15.054002505074937733698562980359, 15.647278006901066148939241708238, 16.25537977326326524564246989868, 17.141797084422141587177281166746, 17.85979809136850956756333996142, 18.40737467458450651650795584899, 19.2215657172160221365009020194, 20.19356608718374182636481274154, 20.98730787759488262609235067593

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