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Properties

Label 1-967-967.122-r0-0-0
Degree $1$
Conductor $967$
Sign $-0.716 + 0.698i$
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  + (0.737 + 0.675i)2-s + (0.993 + 0.116i)3-s + (0.0876 + 0.996i)4-s + (−0.477 + 0.878i)5-s + (0.653 + 0.756i)6-s + (0.938 + 0.344i)7-s + (−0.608 + 0.793i)8-s + (0.972 + 0.232i)9-s + (−0.945 + 0.325i)10-s + (−0.442 + 0.896i)11-s + (−0.0292 + 0.999i)12-s + (−0.442 − 0.896i)13-s + (0.460 + 0.887i)14-s + (−0.576 + 0.816i)15-s + (−0.984 + 0.174i)16-s + (0.924 − 0.380i)17-s + ⋯
L(s)  = 1  + (0.737 + 0.675i)2-s + (0.993 + 0.116i)3-s + (0.0876 + 0.996i)4-s + (−0.477 + 0.878i)5-s + (0.653 + 0.756i)6-s + (0.938 + 0.344i)7-s + (−0.608 + 0.793i)8-s + (0.972 + 0.232i)9-s + (−0.945 + 0.325i)10-s + (−0.442 + 0.896i)11-s + (−0.0292 + 0.999i)12-s + (−0.442 − 0.896i)13-s + (0.460 + 0.887i)14-s + (−0.576 + 0.816i)15-s + (−0.984 + 0.174i)16-s + (0.924 − 0.380i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.716 + 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.716 + 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $-0.716 + 0.698i$
Analytic conductor: \(4.49072\)
Root analytic conductor: \(4.49072\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (122, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (0:\ ),\ -0.716 + 0.698i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.210071429 + 2.974692454i\)
\(L(\frac12)\) \(\approx\) \(1.210071429 + 2.974692454i\)
\(L(1)\) \(\approx\) \(1.572007033 + 1.428713133i\)
\(L(1)\) \(\approx\) \(1.572007033 + 1.428713133i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (0.737 + 0.675i)T \)
3 \( 1 + (0.993 + 0.116i)T \)
5 \( 1 + (-0.477 + 0.878i)T \)
7 \( 1 + (0.938 + 0.344i)T \)
11 \( 1 + (-0.442 + 0.896i)T \)
13 \( 1 + (-0.442 - 0.896i)T \)
17 \( 1 + (0.924 - 0.380i)T \)
19 \( 1 + (0.527 + 0.849i)T \)
23 \( 1 + (0.353 - 0.935i)T \)
29 \( 1 + (0.787 + 0.615i)T \)
31 \( 1 + (0.909 + 0.416i)T \)
37 \( 1 + (-0.977 - 0.212i)T \)
41 \( 1 + (-0.917 - 0.398i)T \)
43 \( 1 + (-0.668 - 0.744i)T \)
47 \( 1 + (-0.998 + 0.0585i)T \)
53 \( 1 + (-0.990 - 0.136i)T \)
59 \( 1 + (-0.511 + 0.859i)T \)
61 \( 1 + (-0.917 - 0.398i)T \)
67 \( 1 + (0.909 - 0.416i)T \)
71 \( 1 + (0.737 - 0.675i)T \)
73 \( 1 + (-0.990 + 0.136i)T \)
79 \( 1 + (0.833 - 0.552i)T \)
83 \( 1 + (0.592 - 0.805i)T \)
89 \( 1 + (0.909 - 0.416i)T \)
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

−21.2779466589767366546863195418, −20.79009128111536595040378004057, −19.98696066817190556233682037127, −19.28318679457403758728912362844, −18.837449646281350654780747380325, −17.62978308293477900397010694439, −16.53221065172923231571485748707, −15.59503208134014699658906139086, −15.00129190165544113997389994162, −13.86915979001700178733636461968, −13.74417874413701988085685586632, −12.74989317096067342742854185487, −11.81060600643298867907395898757, −11.31430930308062370903210606748, −10.08753251081447272274979897715, −9.33885609490935208169028123184, −8.37148952662296993934942629083, −7.74874702452138171256676322669, −6.61061863941488306092488147729, −5.17536117921255623089904162964, −4.68296129430439605465465418714, −3.72047957058351332977258232448, −2.93965889043135212433141173555, −1.71104287633009694800222382967, −1.01613811298212869271590431806, 1.89071810344765870649366576461, 2.90828906621701478877465820133, 3.40929427567100154179199906993, 4.67469318405502051902284126546, 5.17126098458468416879023665943, 6.5888621898075322478905157218, 7.48847020074485317256630661707, 7.9011655239532318796552737742, 8.62175578804611341128534109357, 9.997564881338939437404410235747, 10.66705895768628602279675008566, 12.15188539456281901673980352179, 12.29797483725148404444341276588, 13.65429491154233195454532854135, 14.364620751111064809541652612912, 14.81645578395234987219493759433, 15.41792747598747952161178359430, 16.08901719021342923563865644324, 17.34996665666204370884276000587, 18.196121770917974683648724712627, 18.71198552095774616125946907051, 19.96220713186634605418116889775, 20.65445378224301717218096451193, 21.20585759246643539544533592610, 22.14609405077608441966841269984

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