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Properties

Label 1-967-967.174-r0-0-0
Degree $1$
Conductor $967$
Sign $0.970 - 0.241i$
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  + (0.401 + 0.915i)2-s + (−0.799 + 0.600i)3-s + (−0.677 + 0.735i)4-s + (−0.618 + 0.785i)5-s + (−0.870 − 0.491i)6-s + (0.633 − 0.773i)7-s + (−0.945 − 0.325i)8-s + (0.279 − 0.960i)9-s + (−0.967 − 0.250i)10-s + (0.165 + 0.986i)11-s + (0.100 − 0.994i)12-s + (−0.936 + 0.350i)13-s + (0.962 + 0.269i)14-s + (0.0227 − 0.999i)15-s + (−0.0812 − 0.996i)16-s + (−0.544 + 0.838i)17-s + ⋯
L(s)  = 1  + (0.401 + 0.915i)2-s + (−0.799 + 0.600i)3-s + (−0.677 + 0.735i)4-s + (−0.618 + 0.785i)5-s + (−0.870 − 0.491i)6-s + (0.633 − 0.773i)7-s + (−0.945 − 0.325i)8-s + (0.279 − 0.960i)9-s + (−0.967 − 0.250i)10-s + (0.165 + 0.986i)11-s + (0.100 − 0.994i)12-s + (−0.936 + 0.350i)13-s + (0.962 + 0.269i)14-s + (0.0227 − 0.999i)15-s + (−0.0812 − 0.996i)16-s + (−0.544 + 0.838i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.970 - 0.241i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (174, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.970 - 0.241i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2169110989 + 0.02662687527i\)
\(L(\frac12)\) \(\approx\) \(0.2169110989 + 0.02662687527i\)
\(L(1)\) \(\approx\) \(0.4846699727 + 0.4581821423i\)
\(L(1)\) \(\approx\) \(0.4846699727 + 0.4581821423i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.401 + 0.915i)T \)
3 \( 1 + (-0.799 + 0.600i)T \)
5 \( 1 + (-0.618 + 0.785i)T \)
7 \( 1 + (0.633 - 0.773i)T \)
11 \( 1 + (0.165 + 0.986i)T \)
13 \( 1 + (-0.936 + 0.350i)T \)
17 \( 1 + (-0.544 + 0.838i)T \)
19 \( 1 + (-0.941 + 0.337i)T \)
23 \( 1 + (-0.932 - 0.362i)T \)
29 \( 1 + (0.874 - 0.485i)T \)
31 \( 1 + (0.254 - 0.967i)T \)
37 \( 1 + (-0.132 - 0.991i)T \)
41 \( 1 + (-0.775 + 0.631i)T \)
43 \( 1 + (0.581 - 0.813i)T \)
47 \( 1 + (-0.979 + 0.200i)T \)
53 \( 1 + (-0.974 + 0.225i)T \)
59 \( 1 + (0.436 - 0.899i)T \)
61 \( 1 + (0.934 + 0.356i)T \)
67 \( 1 + (0.710 - 0.703i)T \)
71 \( 1 + (0.592 + 0.805i)T \)
73 \( 1 + (-0.974 - 0.225i)T \)
79 \( 1 + (-0.430 - 0.902i)T \)
83 \( 1 + (0.571 + 0.820i)T \)
89 \( 1 + (-0.964 - 0.263i)T \)
97 \( 1 + (0.623 - 0.781i)T \)
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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

−21.8220819758642416382580704569, −21.15978653910857972588212111656, −20.09226695909552021492601441082, −19.4220261271232268095760037016, −18.85830619831308126808481814767, −17.87199944190525735747916670099, −17.381452008530403837791824973052, −16.2146875669415222426918188737, −15.49292430043694389948855744018, −14.40306376696913657002378627399, −13.55423406997676842966306291386, −12.69961344260919455539489663570, −12.07691827159559648820808351750, −11.56164392695873534947709087017, −10.90854217739144838567117161863, −9.81068939980615197808879095069, −8.62538068953524715088756260715, −8.19917308366867452540648714087, −6.811378891158253326567079073370, −5.74207334993133390774138340730, −4.98764646753189834792261751572, −4.50421480765858081215998367734, −3.05791404111445716146204148918, −2.0439164863961441548785468975, −1.04078259222104734551981890558, 0.10590855223333828732682094138, 2.1969078779632604644635987627, 3.82044266886480916116871206827, 4.257720839987921203505214994282, 4.88416244308926192469043930062, 6.22378288334095229748870778972, 6.75478034215523301034283675155, 7.58851687035161880417639878151, 8.38843526175695646721165258216, 9.761451340215063027582437318315, 10.35600615385181319229865921917, 11.36236708739279045410845734215, 12.10674505710437800497274884023, 12.852674591761841070352584770784, 14.23965561835802774983061848862, 14.692182511433387432668956447244, 15.32492537175567853768055336175, 16.096215589927530825866058734710, 17.148961117961912411992334066502, 17.38801585165899241339073690967, 18.1951583496822765089951151367, 19.286600586639874801769216079838, 20.31630361136398201260159200146, 21.24922412448434735225147172015, 22.02230682197315083749691625033

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