Properties

Label 1-953-953.770-r0-0-0
Degree $1$
Conductor $953$
Sign $-0.744 - 0.667i$
Analytic cond. $4.42571$
Root an. cond. $4.42571$
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  + (−0.982 + 0.183i)2-s + (−0.602 + 0.798i)3-s + (0.932 − 0.361i)4-s + (−0.850 − 0.526i)5-s + (0.445 − 0.895i)6-s + (0.0922 − 0.995i)7-s + (−0.850 + 0.526i)8-s + (−0.273 − 0.961i)9-s + (0.932 + 0.361i)10-s + (0.0922 − 0.995i)11-s + (−0.273 + 0.961i)12-s + (−0.273 − 0.961i)13-s + (0.0922 + 0.995i)14-s + (0.932 − 0.361i)15-s + (0.739 − 0.673i)16-s + (0.445 − 0.895i)17-s + ⋯
L(s)  = 1  + (−0.982 + 0.183i)2-s + (−0.602 + 0.798i)3-s + (0.932 − 0.361i)4-s + (−0.850 − 0.526i)5-s + (0.445 − 0.895i)6-s + (0.0922 − 0.995i)7-s + (−0.850 + 0.526i)8-s + (−0.273 − 0.961i)9-s + (0.932 + 0.361i)10-s + (0.0922 − 0.995i)11-s + (−0.273 + 0.961i)12-s + (−0.273 − 0.961i)13-s + (0.0922 + 0.995i)14-s + (0.932 − 0.361i)15-s + (0.739 − 0.673i)16-s + (0.445 − 0.895i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 953 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.744 - 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 953 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.744 - 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(953\)
Sign: $-0.744 - 0.667i$
Analytic conductor: \(4.42571\)
Root analytic conductor: \(4.42571\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{953} (770, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 953,\ (0:\ ),\ -0.744 - 0.667i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1643452659 - 0.4291684055i\)
\(L(\frac12)\) \(\approx\) \(0.1643452659 - 0.4291684055i\)
\(L(1)\) \(\approx\) \(0.4711582623 - 0.1247176510i\)
\(L(1)\) \(\approx\) \(0.4711582623 - 0.1247176510i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad953 \( 1 \)
good2 \( 1 + (-0.982 + 0.183i)T \)
3 \( 1 + (-0.602 + 0.798i)T \)
5 \( 1 + (-0.850 - 0.526i)T \)
7 \( 1 + (0.0922 - 0.995i)T \)
11 \( 1 + (0.0922 - 0.995i)T \)
13 \( 1 + (-0.273 - 0.961i)T \)
17 \( 1 + (0.445 - 0.895i)T \)
19 \( 1 + (-0.273 - 0.961i)T \)
23 \( 1 + (0.739 + 0.673i)T \)
29 \( 1 + (0.445 - 0.895i)T \)
31 \( 1 + (-0.273 - 0.961i)T \)
37 \( 1 + (-0.982 - 0.183i)T \)
41 \( 1 + (0.932 - 0.361i)T \)
43 \( 1 + (-0.982 - 0.183i)T \)
47 \( 1 + (0.445 + 0.895i)T \)
53 \( 1 + (0.445 - 0.895i)T \)
59 \( 1 + (0.445 - 0.895i)T \)
61 \( 1 + (0.0922 + 0.995i)T \)
67 \( 1 + (-0.273 - 0.961i)T \)
71 \( 1 + (-0.273 - 0.961i)T \)
73 \( 1 + (0.0922 + 0.995i)T \)
79 \( 1 + (-0.273 + 0.961i)T \)
83 \( 1 + (0.0922 + 0.995i)T \)
89 \( 1 + (-0.850 + 0.526i)T \)
97 \( 1 + (0.0922 - 0.995i)T \)
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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

−21.996862133801806842434390876677, −21.35241224312808454794821170471, −20.1851901393273816059975129501, −19.34837676258225739599539184564, −18.8809385648874092531579158163, −18.32439836652656770867552210168, −17.56774635025850597264904518774, −16.67783704101639892226866770405, −16.04321260125755462412297420665, −14.968547865391206101425794070994, −14.4732391916209352027808914583, −12.70787597882634883640979864713, −12.2027335916246138521022900868, −11.78729743425724962103129831075, −10.78434607303393471355915666085, −10.15495314585259184057999979877, −8.820875466759827818348394467234, −8.26238993472549837727170709272, −7.22184789351250032886163912066, −6.79270945770617255802769814705, −5.845617618261774763356245210, −4.5761918590834074718312939152, −3.21226580430994690774293949082, −2.16069706497970272039296599355, −1.43766017812963646713828271156, 0.413753114503570826144332887171, 0.90161223458780248548252793965, 2.93302461957765864664943941106, 3.76486567022408501439102027650, 4.91654499680880470355608689743, 5.65155131461519356731793203928, 6.85125975161017767775230219785, 7.618672446090748699277967252887, 8.46700869289601654170866195187, 9.33641340139933339788529733781, 10.11055330615105736306225577291, 11.133529248702666737496206608800, 11.286315699741024278136937619562, 12.33975306672586991894697633159, 13.52418351327719501122726704822, 14.71856962657293234881664294832, 15.563023002167437658672586561776, 16.0114077714218799507560048466, 16.916571268194739519606503550209, 17.20427533214772534841370964994, 18.15972452081440237554973260377, 19.33529442739332801602545903206, 19.759331525166576601657040499520, 20.76978068958985288313066514301, 21.02813715009030083083160386499

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