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Properties

Label 1-967-967.461-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: $\chi_{967} (461, \cdot )$
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

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

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