Properties

Label 1-967-967.458-r0-0-0
Degree $1$
Conductor $967$
Sign $-0.799 - 0.600i$
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  + (0.100 − 0.994i)2-s + (0.909 + 0.416i)3-s + (−0.979 − 0.200i)4-s + (0.701 − 0.712i)5-s + (0.505 − 0.862i)6-s + (−0.971 − 0.238i)7-s + (−0.297 + 0.954i)8-s + (0.653 + 0.756i)9-s + (−0.638 − 0.769i)10-s + (0.592 − 0.805i)11-s + (−0.807 − 0.589i)12-s + (−0.993 + 0.110i)13-s + (−0.334 + 0.942i)14-s + (0.934 − 0.356i)15-s + (0.919 + 0.392i)16-s + (0.787 + 0.615i)17-s + ⋯
L(s)  = 1  + (0.100 − 0.994i)2-s + (0.909 + 0.416i)3-s + (−0.979 − 0.200i)4-s + (0.701 − 0.712i)5-s + (0.505 − 0.862i)6-s + (−0.971 − 0.238i)7-s + (−0.297 + 0.954i)8-s + (0.653 + 0.756i)9-s + (−0.638 − 0.769i)10-s + (0.592 − 0.805i)11-s + (−0.807 − 0.589i)12-s + (−0.993 + 0.110i)13-s + (−0.334 + 0.942i)14-s + (0.934 − 0.356i)15-s + (0.919 + 0.392i)16-s + (0.787 + 0.615i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.799 - 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.799 - 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.799 - 0.600i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (458, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ -0.799 - 0.600i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5353253641 - 1.605332163i\)
\(L(\frac12)\) \(\approx\) \(0.5353253641 - 1.605332163i\)
\(L(1)\) \(\approx\) \(1.036585833 - 0.7937009352i\)
\(L(1)\) \(\approx\) \(1.036585833 - 0.7937009352i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.100 - 0.994i)T \)
3 \( 1 + (0.909 + 0.416i)T \)
5 \( 1 + (0.701 - 0.712i)T \)
7 \( 1 + (-0.971 - 0.238i)T \)
11 \( 1 + (0.592 - 0.805i)T \)
13 \( 1 + (-0.993 + 0.110i)T \)
17 \( 1 + (0.787 + 0.615i)T \)
19 \( 1 + (-0.837 - 0.547i)T \)
23 \( 1 + (-0.696 - 0.717i)T \)
29 \( 1 + (-0.184 - 0.982i)T \)
31 \( 1 + (0.867 - 0.497i)T \)
37 \( 1 + (-0.705 - 0.708i)T \)
41 \( 1 + (-0.0682 - 0.997i)T \)
43 \( 1 + (0.997 - 0.0649i)T \)
47 \( 1 + (0.304 - 0.952i)T \)
53 \( 1 + (-0.877 - 0.480i)T \)
59 \( 1 + (-0.922 - 0.386i)T \)
61 \( 1 + (0.898 + 0.439i)T \)
67 \( 1 + (-0.864 + 0.502i)T \)
71 \( 1 + (0.811 - 0.584i)T \)
73 \( 1 + (-0.877 + 0.480i)T \)
79 \( 1 + (0.998 - 0.0520i)T \)
83 \( 1 + (0.728 + 0.684i)T \)
89 \( 1 + (-0.00325 - 0.999i)T \)
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

−22.17228505970744522368982489278, −21.52545062929281436891166431493, −20.46613483047060731644836933183, −19.314968519201380457175160975644, −19.00194691425910182507914073596, −18.02488037064449601546289204457, −17.42990992778214386575176342192, −16.50880297832561179935435817694, −15.48484985273150776350428546043, −14.84177348690545226775745994356, −14.212059365800564071698739607187, −13.63743640777142402142959088830, −12.56133342838303883888103832159, −12.23689519202155777610674707281, −10.22459783248887734490350305906, −9.64577714474204052509367739877, −9.18388705968509367147801318495, −7.95554897407950301770932348568, −7.18392871630110770767645840337, −6.60310281896830438217884362570, −5.83898397424131475855814311695, −4.60338025536386403428642465424, −3.447395842105557116219916454915, −2.764203857942582184161066208819, −1.52446504504637334519276574595, 0.61874825470614211494365749849, 1.99413023084669188827917769237, 2.649316902542718416725001475602, 3.76116514177116528211365244580, 4.3276699742780908948438015661, 5.448010098049879028500672419059, 6.408026784687035438745830951905, 7.90917789851713251265830476463, 8.783256657531495813497592163551, 9.35704965695838584164637859969, 10.06712151961167299316017806984, 10.6136954482130392071411921171, 12.05644243513915050786576005936, 12.67390526892571174634672528599, 13.468564206567057911554522022849, 14.029235337323499495075743115949, 14.776509428409655965713294764750, 15.93501676844703915824883093965, 16.90118164469410224554474027085, 17.33509115673920962886988222766, 18.83992617166488100450170802046, 19.30911966952658217112713019551, 19.81628084796213922802208400926, 20.742110450674697911927808838398, 21.26982150897129008590763019460

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