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Properties

Label 1-967-967.70-r0-0-0
Degree $1$
Conductor $967$
Sign $0.104 + 0.994i$
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  + (0.701 + 0.712i)2-s + (−0.932 + 0.362i)3-s + (−0.0162 + 0.999i)4-s + (−0.964 + 0.263i)5-s + (−0.911 − 0.410i)6-s + (−0.555 − 0.831i)7-s + (−0.724 + 0.689i)8-s + (0.737 − 0.675i)9-s + (−0.864 − 0.502i)10-s + (−0.371 + 0.928i)11-s + (−0.347 − 0.937i)12-s + (0.989 + 0.142i)13-s + (0.203 − 0.979i)14-s + (0.803 − 0.595i)15-s + (−0.999 − 0.0325i)16-s + (0.653 − 0.756i)17-s + ⋯
L(s)  = 1  + (0.701 + 0.712i)2-s + (−0.932 + 0.362i)3-s + (−0.0162 + 0.999i)4-s + (−0.964 + 0.263i)5-s + (−0.911 − 0.410i)6-s + (−0.555 − 0.831i)7-s + (−0.724 + 0.689i)8-s + (0.737 − 0.675i)9-s + (−0.864 − 0.502i)10-s + (−0.371 + 0.928i)11-s + (−0.347 − 0.937i)12-s + (0.989 + 0.142i)13-s + (0.203 − 0.979i)14-s + (0.803 − 0.595i)15-s + (−0.999 − 0.0325i)16-s + (0.653 − 0.756i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.104 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.104 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.104 + 0.994i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (70, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ 0.104 + 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8385045523 + 0.7551020161i\)
\(L(\frac12)\) \(\approx\) \(0.8385045523 + 0.7551020161i\)
\(L(1)\) \(\approx\) \(0.8125387274 + 0.4753860372i\)
\(L(1)\) \(\approx\) \(0.8125387274 + 0.4753860372i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.701 + 0.712i)T \)
3 \( 1 + (-0.932 + 0.362i)T \)
5 \( 1 + (-0.964 + 0.263i)T \)
7 \( 1 + (-0.555 - 0.831i)T \)
11 \( 1 + (-0.371 + 0.928i)T \)
13 \( 1 + (0.989 + 0.142i)T \)
17 \( 1 + (0.653 - 0.756i)T \)
19 \( 1 + (0.436 - 0.899i)T \)
23 \( 1 + (0.165 - 0.986i)T \)
29 \( 1 + (0.494 - 0.869i)T \)
31 \( 1 + (-0.209 + 0.977i)T \)
37 \( 1 + (-0.359 + 0.933i)T \)
41 \( 1 + (0.962 + 0.269i)T \)
43 \( 1 + (-0.533 - 0.845i)T \)
47 \( 1 + (-0.759 - 0.651i)T \)
53 \( 1 + (0.995 + 0.0909i)T \)
59 \( 1 + (-0.132 + 0.991i)T \)
61 \( 1 + (-0.715 + 0.699i)T \)
67 \( 1 + (0.951 + 0.307i)T \)
71 \( 1 + (-0.967 - 0.250i)T \)
73 \( 1 + (0.995 - 0.0909i)T \)
79 \( 1 + (0.692 + 0.721i)T \)
83 \( 1 + (0.643 + 0.765i)T \)
89 \( 1 + (-0.741 + 0.670i)T \)
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

−21.516255610995466646721817455400, −21.108000515160712837837889756513, −19.88780131145351911540926582903, −19.10058685232176394289734001911, −18.729767362686886105090007439509, −17.973095732567753954392217086481, −16.498535594205068091589028798512, −16.01444331601105594157794375217, −15.37973414226678083226748118435, −14.27407522552400435008291963081, −13.15807022407993766796854752482, −12.689801964776345672674931060540, −11.97398544959920230221711540397, −11.26919796917532117861389499674, −10.7089844998467745706965505711, −9.6292563055255078019108371589, −8.51971144361099895086804866539, −7.58749485297768116291912472723, −6.2434250978149773990739443760, −5.78758888148909365943775348350, −5.01752353645912411664029115884, −3.73440907765953164622958717073, −3.24514437398288361181610690422, −1.73523042089116716373279571484, −0.73163039955201114439091367960, 0.73896714939167728388069919420, 2.8893239708116910951778180130, 3.766675620725222861130615572079, 4.495814506607110934008513311726, 5.16948057166313745900713678983, 6.44151026123499462844019375727, 6.95120518566324692646021339578, 7.6202776415041695286908334504, 8.77678410580914600889052898799, 9.98084090041011863160617912011, 10.80794538778890856924499930715, 11.68429025930592578959923451635, 12.29628150062778744426269509185, 13.15456840798424793360487053016, 13.99427369095499480280377373472, 15.11784561573326338952688103297, 15.64845945707160218466528699440, 16.317238402586401082375418437444, 16.83056341372709073281514029908, 17.92892813396112950694778764434, 18.42937976057470996049647454750, 19.76447559752640127618601194197, 20.616753179450232488989497445391, 21.21542052689864785491444254301, 22.52757283402561207269546901024

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