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Properties

Label 1-967-967.136-r0-0-0
Degree $1$
Conductor $967$
Sign $-0.734 - 0.678i$
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  + (0.279 + 0.960i)2-s + (−0.957 + 0.288i)3-s + (−0.844 + 0.536i)4-s + (−0.442 − 0.896i)5-s + (−0.544 − 0.838i)6-s + (−0.638 + 0.769i)7-s + (−0.750 − 0.660i)8-s + (0.833 − 0.552i)9-s + (0.737 − 0.675i)10-s + (0.353 + 0.935i)11-s + (0.653 − 0.756i)12-s + (0.353 − 0.935i)13-s + (−0.917 − 0.398i)14-s + (0.682 + 0.730i)15-s + (0.425 − 0.905i)16-s + (0.560 + 0.828i)17-s + ⋯
L(s)  = 1  + (0.279 + 0.960i)2-s + (−0.957 + 0.288i)3-s + (−0.844 + 0.536i)4-s + (−0.442 − 0.896i)5-s + (−0.544 − 0.838i)6-s + (−0.638 + 0.769i)7-s + (−0.750 − 0.660i)8-s + (0.833 − 0.552i)9-s + (0.737 − 0.675i)10-s + (0.353 + 0.935i)11-s + (0.653 − 0.756i)12-s + (0.353 − 0.935i)13-s + (−0.917 − 0.398i)14-s + (0.682 + 0.730i)15-s + (0.425 − 0.905i)16-s + (0.560 + 0.828i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.734 - 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.734 - 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.734 - 0.678i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ -0.734 - 0.678i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1254104201 + 0.3203901622i\)
\(L(\frac12)\) \(\approx\) \(-0.1254104201 + 0.3203901622i\)
\(L(1)\) \(\approx\) \(0.5041615187 + 0.3776991623i\)
\(L(1)\) \(\approx\) \(0.5041615187 + 0.3776991623i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.279 + 0.960i)T \)
3 \( 1 + (-0.957 + 0.288i)T \)
5 \( 1 + (-0.442 - 0.896i)T \)
7 \( 1 + (-0.638 + 0.769i)T \)
11 \( 1 + (0.353 + 0.935i)T \)
13 \( 1 + (0.353 - 0.935i)T \)
17 \( 1 + (0.560 + 0.828i)T \)
19 \( 1 + (-0.822 - 0.568i)T \)
23 \( 1 + (0.993 - 0.116i)T \)
29 \( 1 + (0.0876 + 0.996i)T \)
31 \( 1 + (-0.477 + 0.878i)T \)
37 \( 1 + (-0.511 - 0.859i)T \)
41 \( 1 + (0.854 + 0.519i)T \)
43 \( 1 + (-0.864 + 0.502i)T \)
47 \( 1 + (-0.145 + 0.989i)T \)
53 \( 1 + (-0.334 - 0.942i)T \)
59 \( 1 + (0.527 + 0.849i)T \)
61 \( 1 + (0.854 + 0.519i)T \)
67 \( 1 + (-0.477 - 0.878i)T \)
71 \( 1 + (0.279 - 0.960i)T \)
73 \( 1 + (-0.334 + 0.942i)T \)
79 \( 1 + (-0.107 - 0.994i)T \)
83 \( 1 + (-0.696 + 0.717i)T \)
89 \( 1 + (-0.477 - 0.878i)T \)
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

−21.40511262400369236295324998670, −20.584355365109717579560754308923, −19.38030029621206197888784387904, −18.83613093309970358045931647421, −18.66613880031726434135863191555, −17.32990833254412873659194013020, −16.71647774489657855934575305920, −15.82008970341181933412921530144, −14.67361514266903272434162836684, −13.78556463772017799274001702471, −13.29771267405275765839400197691, −12.20053780706941293998879717613, −11.48973493339264750248580852251, −11.01040974246759231628932876636, −10.24615253468027653282712207956, −9.47093561219477295098952216988, −8.175000264759348453752743972083, −6.988000821985691727190948799033, −6.38705409358905406675499434031, −5.47680715062945386465600938965, −4.18122849843266391172705306516, −3.69361931817608378447143561590, −2.58242519683671852912021073788, −1.285049017245612072856237339492, −0.18470176675930847933801442261, 1.26325036061695204105700624324, 3.22632938421035225614561026161, 4.155859289641348990148335940945, 5.02046888660165274393189231559, 5.58626875702080860949107691666, 6.49014492816848858382080675757, 7.27375795612405015750430030672, 8.43278897189186228970287104655, 9.09901514287091675686974789488, 9.935641712621675782285986656618, 11.054495298888238916466191864, 12.3077677406295044413940508199, 12.62542474324127263628803123585, 13.081499920478872875424057347863, 14.86615214778104130243857870787, 15.12757128003391591301296724902, 16.11903613988926369409247143527, 16.43554939835511325324901377805, 17.48777693449181194315906485102, 17.814644636764361284492526049805, 18.96056067301002534672605932719, 19.81249422536449434831885020050, 21.07344749437405132132793022618, 21.550697227278790288331620595105, 22.58090784955350345076789906906

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