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Properties

Label 1-967-967.703-r0-0-0
Degree $1$
Conductor $967$
Sign $-0.997 - 0.0684i$
Analytic cond. $4.49072$
Root an. cond. $4.49072$
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  + (0.962 − 0.269i)2-s + (0.203 − 0.979i)3-s + (0.854 − 0.519i)4-s + (0.460 − 0.887i)5-s + (−0.0682 − 0.997i)6-s + (−0.576 + 0.816i)7-s + (0.682 − 0.730i)8-s + (−0.917 − 0.398i)9-s + (0.203 − 0.979i)10-s + (−0.917 − 0.398i)11-s + (−0.334 − 0.942i)12-s + (−0.917 + 0.398i)13-s + (−0.334 + 0.942i)14-s + (−0.775 − 0.631i)15-s + (0.460 − 0.887i)16-s + (−0.775 + 0.631i)17-s + ⋯
L(s)  = 1  + (0.962 − 0.269i)2-s + (0.203 − 0.979i)3-s + (0.854 − 0.519i)4-s + (0.460 − 0.887i)5-s + (−0.0682 − 0.997i)6-s + (−0.576 + 0.816i)7-s + (0.682 − 0.730i)8-s + (−0.917 − 0.398i)9-s + (0.203 − 0.979i)10-s + (−0.917 − 0.398i)11-s + (−0.334 − 0.942i)12-s + (−0.917 + 0.398i)13-s + (−0.334 + 0.942i)14-s + (−0.775 − 0.631i)15-s + (0.460 − 0.887i)16-s + (−0.775 + 0.631i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.997 - 0.0684i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ -0.997 - 0.0684i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07237183881 - 2.110895410i\)
\(L(\frac12)\) \(\approx\) \(0.07237183881 - 2.110895410i\)
\(L(1)\) \(\approx\) \(1.211025080 - 1.145482689i\)
\(L(1)\) \(\approx\) \(1.211025080 - 1.145482689i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.962 - 0.269i)T \)
3 \( 1 + (0.203 - 0.979i)T \)
5 \( 1 + (0.460 - 0.887i)T \)
7 \( 1 + (-0.576 + 0.816i)T \)
11 \( 1 + (-0.917 - 0.398i)T \)
13 \( 1 + (-0.917 + 0.398i)T \)
17 \( 1 + (-0.775 + 0.631i)T \)
19 \( 1 + (0.203 - 0.979i)T \)
23 \( 1 + (0.854 - 0.519i)T \)
29 \( 1 + (-0.917 + 0.398i)T \)
31 \( 1 + (0.682 - 0.730i)T \)
37 \( 1 + (-0.917 + 0.398i)T \)
41 \( 1 + (-0.0682 - 0.997i)T \)
43 \( 1 + (-0.775 - 0.631i)T \)
47 \( 1 + (-0.775 - 0.631i)T \)
53 \( 1 + (0.854 - 0.519i)T \)
59 \( 1 + (0.854 + 0.519i)T \)
61 \( 1 + (-0.0682 - 0.997i)T \)
67 \( 1 + (0.682 + 0.730i)T \)
71 \( 1 + (0.962 + 0.269i)T \)
73 \( 1 + (0.854 + 0.519i)T \)
79 \( 1 + (0.854 + 0.519i)T \)
83 \( 1 + (-0.0682 - 0.997i)T \)
89 \( 1 + (0.682 + 0.730i)T \)
97 \( 1 + T \)
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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

−22.31379812119014928130722765374, −21.35081242351899920753202828355, −20.86410037799697038777187083719, −20.015948770874612844985970372187, −19.33883766045523475167835702043, −17.96699146015684508160029428528, −17.15150681875498777379493605668, −16.434980739842361804967263121827, −15.5531432418150159071303461709, −15.02821758172633018867483416702, −14.23862432296183330029601514759, −13.55465251716775702014643576822, −12.85195521251187228703122979460, −11.61430310778592084953664150386, −10.768527644558594792417286420998, −10.1789338281544561363475213252, −9.489706484631818109112323587209, −7.94732658237737060371134587965, −7.262514416999327789210973036414, −6.3987502872752072094858657777, −5.33612536040696668165661561209, −4.729792825152828733379758962583, −3.5704564315922884361744875335, −3.00655850044661473352217911513, −2.15051528258202014377669890789, 0.551541223748226374891493105445, 2.097973487673026291711398261714, 2.36737280667299377634004845883, 3.51789207166036700130045487021, 4.99631239765483646318695872785, 5.40960456552082829464371502962, 6.442123968503665950279952870212, 7.0745406027188939908693390396, 8.34979754795176047059279556326, 9.06979028067158302076628944684, 10.075589081019038828837801510086, 11.27902743315849386075522959185, 12.03862905892931185473721751680, 12.83595595685791971096102271688, 13.147760494277530680125994721237, 13.85178575019942236672567051823, 14.96882142855924621136162915170, 15.574201957748981988415156954270, 16.62549311654975571115049436552, 17.35488829852021808013137687581, 18.50156438708526944681319073312, 19.164615028708623851478221241023, 19.84692856389972000347275543118, 20.60861324976795643995643084800, 21.43088957757398893544136027709

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