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Properties

Label 1-967-967.152-r1-0-0
Degree $1$
Conductor $967$
Sign $0.933 - 0.358i$
Analytic cond. $103.918$
Root an. cond. $103.918$
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  + (−0.677 − 0.735i)2-s + (−0.279 − 0.960i)3-s + (−0.0812 + 0.996i)4-s + (0.235 − 0.971i)5-s + (−0.516 + 0.856i)6-s + (0.197 − 0.980i)7-s + (0.787 − 0.615i)8-s + (−0.844 + 0.536i)9-s + (−0.874 + 0.485i)10-s + (−0.945 − 0.325i)11-s + (0.979 − 0.200i)12-s + (−0.754 − 0.655i)13-s + (−0.854 + 0.519i)14-s + (−0.998 + 0.0455i)15-s + (−0.986 − 0.161i)16-s + (−0.407 + 0.913i)17-s + ⋯
L(s)  = 1  + (−0.677 − 0.735i)2-s + (−0.279 − 0.960i)3-s + (−0.0812 + 0.996i)4-s + (0.235 − 0.971i)5-s + (−0.516 + 0.856i)6-s + (0.197 − 0.980i)7-s + (0.787 − 0.615i)8-s + (−0.844 + 0.536i)9-s + (−0.874 + 0.485i)10-s + (−0.945 − 0.325i)11-s + (0.979 − 0.200i)12-s + (−0.754 − 0.655i)13-s + (−0.854 + 0.519i)14-s + (−0.998 + 0.0455i)15-s + (−0.986 − 0.161i)16-s + (−0.407 + 0.913i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.933 - 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.933 - 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.933 - 0.358i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (152, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ 0.933 - 0.358i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1256828867 + 0.02326955921i\)
\(L(\frac12)\) \(\approx\) \(-0.1256828867 + 0.02326955921i\)
\(L(1)\) \(\approx\) \(0.2685593745 - 0.4350156204i\)
\(L(1)\) \(\approx\) \(0.2685593745 - 0.4350156204i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.677 - 0.735i)T \)
3 \( 1 + (-0.279 - 0.960i)T \)
5 \( 1 + (0.235 - 0.971i)T \)
7 \( 1 + (0.197 - 0.980i)T \)
11 \( 1 + (-0.945 - 0.325i)T \)
13 \( 1 + (-0.754 - 0.655i)T \)
17 \( 1 + (-0.407 + 0.913i)T \)
19 \( 1 + (-0.771 - 0.636i)T \)
23 \( 1 + (-0.737 + 0.675i)T \)
29 \( 1 + (-0.527 - 0.849i)T \)
31 \( 1 + (-0.870 + 0.491i)T \)
37 \( 1 + (0.964 + 0.263i)T \)
41 \( 1 + (-0.203 - 0.979i)T \)
43 \( 1 + (0.322 - 0.946i)T \)
47 \( 1 + (-0.919 - 0.392i)T \)
53 \( 1 + (0.898 + 0.439i)T \)
59 \( 1 + (-0.618 + 0.785i)T \)
61 \( 1 + (0.746 - 0.665i)T \)
67 \( 1 + (-0.00975 - 0.999i)T \)
71 \( 1 + (-0.297 - 0.954i)T \)
73 \( 1 + (0.898 - 0.439i)T \)
79 \( 1 + (0.628 + 0.777i)T \)
83 \( 1 + (-0.347 - 0.937i)T \)
89 \( 1 + (-0.861 + 0.508i)T \)
97 \( 1 + (-0.222 + 0.974i)T \)
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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

−22.36145553252186613746608472977, −21.75550668967721077259410786782, −20.89782825061121669555665824891, −19.96019444528544399415904835800, −18.9426659913932844035617603818, −18.14398683765753396618596341272, −17.9128453373025663875485958712, −16.65870538612991890970437963117, −16.15485496240719363098670520079, −15.247778571377477736145416803091, −14.69283980456133678698500091954, −14.26652815851312385974881998014, −12.817392904176047785524947827295, −11.49895628397436769419682426322, −11.0177855467026575994003629277, −9.96442204773850546435309564960, −9.62578957342928035548214261603, −8.64444462163882569205974295757, −7.71698582865799002998755749368, −6.70661905550887924322389515924, −5.90688317775658788943799704441, −5.16627082598571905537586320364, −4.29898201103666994311336389923, −2.72256117591612873994529521783, −2.06756148122723095169795461548, 0.06099832900663032023254401231, 0.535811577993903935778055700611, 1.696686339686683282220865572161, 2.36985228735899482012903626282, 3.73088371618324627714214375433, 4.784894293709503770403038922955, 5.7537695056207849515560549787, 7.00180362557177838261057482308, 7.8849619153993051264291559628, 8.24482220614574050442885578409, 9.3266665501235476423443041971, 10.40512825625524163188613779990, 10.91769987206903300758731990041, 11.94774572950538170494719738168, 12.74336583141275111219871013110, 13.23522782806187976851294500377, 13.79746333633941541410877135792, 15.313426359304565995607079189091, 16.50230157697258830929212010810, 17.06217165294461906621238508724, 17.597842671013994187859330703137, 18.24328681274128659328655590402, 19.39864317983316574684859805483, 19.8026000720576235386488099130, 20.450983029362454686751363331463

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