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Properties

Label 1-967-967.881-r0-0-0
Degree $1$
Conductor $967$
Sign $0.893 - 0.448i$
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  + (0.795 − 0.605i)2-s + (−0.184 − 0.982i)3-s + (0.266 − 0.963i)4-s + (−0.791 + 0.610i)5-s + (−0.741 − 0.670i)6-s + (−0.999 + 0.0325i)7-s + (−0.371 − 0.928i)8-s + (−0.932 + 0.362i)9-s + (−0.260 + 0.965i)10-s + (0.560 + 0.828i)11-s + (−0.996 − 0.0844i)12-s + (0.436 + 0.899i)13-s + (−0.775 + 0.631i)14-s + (0.746 + 0.665i)15-s + (−0.857 − 0.514i)16-s + (0.909 − 0.416i)17-s + ⋯
L(s)  = 1  + (0.795 − 0.605i)2-s + (−0.184 − 0.982i)3-s + (0.266 − 0.963i)4-s + (−0.791 + 0.610i)5-s + (−0.741 − 0.670i)6-s + (−0.999 + 0.0325i)7-s + (−0.371 − 0.928i)8-s + (−0.932 + 0.362i)9-s + (−0.260 + 0.965i)10-s + (0.560 + 0.828i)11-s + (−0.996 − 0.0844i)12-s + (0.436 + 0.899i)13-s + (−0.775 + 0.631i)14-s + (0.746 + 0.665i)15-s + (−0.857 − 0.514i)16-s + (0.909 − 0.416i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.893 - 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.893 - 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.893 - 0.448i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.893 - 0.448i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.395043616 - 0.3307493761i\)
\(L(\frac12)\) \(\approx\) \(1.395043616 - 0.3307493761i\)
\(L(1)\) \(\approx\) \(1.081927463 - 0.4895545792i\)
\(L(1)\) \(\approx\) \(1.081927463 - 0.4895545792i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.795 - 0.605i)T \)
3 \( 1 + (-0.184 - 0.982i)T \)
5 \( 1 + (-0.791 + 0.610i)T \)
7 \( 1 + (-0.999 + 0.0325i)T \)
11 \( 1 + (0.560 + 0.828i)T \)
13 \( 1 + (0.436 + 0.899i)T \)
17 \( 1 + (0.909 - 0.416i)T \)
19 \( 1 + (0.0357 + 0.999i)T \)
23 \( 1 + (0.763 - 0.646i)T \)
29 \( 1 + (-0.864 + 0.502i)T \)
31 \( 1 + (-0.359 + 0.933i)T \)
37 \( 1 + (-0.430 + 0.902i)T \)
41 \( 1 + (-0.990 - 0.136i)T \)
43 \( 1 + (0.516 + 0.856i)T \)
47 \( 1 + (0.985 - 0.168i)T \)
53 \( 1 + (0.538 - 0.842i)T \)
59 \( 1 + (-0.322 + 0.946i)T \)
61 \( 1 + (0.613 - 0.789i)T \)
67 \( 1 + (0.987 + 0.155i)T \)
71 \( 1 + (0.126 - 0.991i)T \)
73 \( 1 + (0.538 + 0.842i)T \)
79 \( 1 + (-0.120 - 0.992i)T \)
83 \( 1 + (0.818 - 0.574i)T \)
89 \( 1 + (-0.628 + 0.777i)T \)
97 \( 1 + (-0.900 + 0.433i)T \)
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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

−22.021126484762557501310109703, −21.16352887508180579381318830212, −20.40589079386136995582199163831, −19.73884181619975483796517802152, −18.81539445302318622966988296569, −17.19104181014607002574149007752, −16.91803514983194727666719676883, −16.09509165194800824023894005160, −15.48165118029240429219439075214, −15.04797773565758828874216764032, −13.857310645975905068029348083075, −13.08055342897952216319667833631, −12.29980499753343949399111756443, −11.432576537499565155174940718554, −10.768785542202338936005653223222, −9.40323943090263988547463407467, −8.80126694388947329138885041179, −7.894489300387235034423301749732, −6.877864039950453545413313582978, −5.66891996760753548172515564489, −5.43260278665688889635819783760, −4.015799289201273126555880911474, −3.66976884390175922433843131363, −2.8881584101945093797422373292, −0.56635237765891825003275964289, 1.08703650941113472170391920540, 2.12440572201310109816875520348, 3.20644706467386508316995972780, 3.77550619233903184728950134125, 5.01479113693515035817455951839, 6.136487293665439066893386175918, 6.82843801081724660768154567845, 7.29574498505513362428520947155, 8.69919281758649493984501535975, 9.75535841275145822789772119741, 10.662425475678366286033911224681, 11.54876991454187348895960184694, 12.2391114621622518303857929864, 12.60365440815167342727090367665, 13.6889689797248540791258283404, 14.387246495248800500826270596649, 15.023946350114531256816528633485, 16.17313195903880051300287238511, 16.78041145772729044036854647161, 18.28431442165212029743438785734, 18.79819300201373412161579108483, 19.24346126463082675629756477369, 20.06360925746909100133727630580, 20.73549054247710049175948292175, 22.0309993658333479233737890271

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