Properties

Label 1-967-967.27-r1-0-0
Degree $1$
Conductor $967$
Sign $0.788 + 0.614i$
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  + (−0.145 + 0.989i)2-s + (−0.924 + 0.380i)3-s + (−0.957 − 0.288i)4-s + (−0.0876 − 0.996i)5-s + (−0.241 − 0.970i)6-s + (−0.389 + 0.921i)7-s + (0.425 − 0.905i)8-s + (0.710 − 0.703i)9-s + (0.998 + 0.0585i)10-s + (−0.844 − 0.536i)11-s + (0.995 − 0.0974i)12-s + (0.844 − 0.536i)13-s + (−0.854 − 0.519i)14-s + (0.460 + 0.887i)15-s + (0.833 + 0.552i)16-s + (−0.967 − 0.250i)17-s + ⋯
L(s)  = 1  + (−0.145 + 0.989i)2-s + (−0.924 + 0.380i)3-s + (−0.957 − 0.288i)4-s + (−0.0876 − 0.996i)5-s + (−0.241 − 0.970i)6-s + (−0.389 + 0.921i)7-s + (0.425 − 0.905i)8-s + (0.710 − 0.703i)9-s + (0.998 + 0.0585i)10-s + (−0.844 − 0.536i)11-s + (0.995 − 0.0974i)12-s + (0.844 − 0.536i)13-s + (−0.854 − 0.519i)14-s + (0.460 + 0.887i)15-s + (0.833 + 0.552i)16-s + (−0.967 − 0.250i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.788 + 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.788 + 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(967\)
Sign: $0.788 + 0.614i$
Analytic conductor: \(103.918\)
Root analytic conductor: \(103.918\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{967} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 967,\ (1:\ ),\ 0.788 + 0.614i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4109709188 + 0.1412733910i\)
\(L(\frac12)\) \(\approx\) \(0.4109709188 + 0.1412733910i\)
\(L(1)\) \(\approx\) \(0.4601366722 + 0.2192933402i\)
\(L(1)\) \(\approx\) \(0.4601366722 + 0.2192933402i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad967 \( 1 \)
good2 \( 1 + (-0.145 + 0.989i)T \)
3 \( 1 + (-0.924 + 0.380i)T \)
5 \( 1 + (-0.0876 - 0.996i)T \)
7 \( 1 + (-0.389 + 0.921i)T \)
11 \( 1 + (-0.844 - 0.536i)T \)
13 \( 1 + (0.844 - 0.536i)T \)
17 \( 1 + (-0.967 - 0.250i)T \)
19 \( 1 + (-0.279 + 0.960i)T \)
23 \( 1 + (-0.987 + 0.155i)T \)
29 \( 1 + (-0.993 + 0.116i)T \)
31 \( 1 + (0.787 + 0.615i)T \)
37 \( 1 + (0.945 + 0.325i)T \)
41 \( 1 + (-0.203 + 0.979i)T \)
43 \( 1 + (-0.763 + 0.646i)T \)
47 \( 1 + (-0.981 - 0.193i)T \)
53 \( 1 + (-0.0682 + 0.997i)T \)
59 \( 1 + (0.737 - 0.675i)T \)
61 \( 1 + (0.203 - 0.979i)T \)
67 \( 1 + (-0.787 + 0.615i)T \)
71 \( 1 + (-0.145 - 0.989i)T \)
73 \( 1 + (-0.0682 - 0.997i)T \)
79 \( 1 + (0.371 - 0.928i)T \)
83 \( 1 + (-0.999 + 0.0195i)T \)
89 \( 1 + (-0.787 + 0.615i)T \)
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

−21.55492894867180743943920611555, −20.69526781477536398346661268103, −19.747370033784884242600925356796, −19.10528705951359000480655649573, −18.247031711972798391173801435002, −17.86235073381221700745063960587, −17.05146237940641571977034899893, −16.09240497955612559249556856309, −15.15422014055866735642712633529, −13.83195583305729885426299102029, −13.347455796800689774586503532517, −12.69698399930067464189430500947, −11.340796910622794702867863062914, −11.26220170612979673720206270873, −10.28598886109707635916296112211, −9.84499553288481589170661321524, −8.41046844517784313499755092439, −7.394937910560763049828747851251, −6.73108830069895349762762335040, −5.7464774928690677676549826606, −4.43560461365509124087854878435, −3.91246343421104337481703670410, −2.57073703535837620522876081946, −1.78571424171994761235670567637, −0.39746431401813361846833316817, 0.281115983998198561207690549074, 1.51896366096077029675093548526, 3.39476619004800293106044038241, 4.45491931418620701735828878725, 5.22832286562440304283397255447, 5.931950477212254252691213205092, 6.4023132896623584075943451582, 7.93157784754534314384228170295, 8.458007156026837930921824936954, 9.39540194139098905627360736151, 10.087783652193575144050193605431, 11.162020599598560300329668275202, 12.150131708458225493186409156352, 13.00087858202873035044850528620, 13.4352434822620170015469424225, 14.939679342459817496191759802068, 15.607038135991256308031585539844, 16.22390818958601636252855771333, 16.52686308702197374874195931628, 17.71210645576810592497830549885, 18.191081108338245368789726247438, 18.9047317997064520744086456873, 20.110459402560673931875472828302, 21.11132314784704766684886110980, 21.779938001148586789162864983780

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