Properties

Label 1-1183-1183.298-r0-0-0
Degree $1$
Conductor $1183$
Sign $0.915 - 0.401i$
Analytic cond. $5.49382$
Root an. cond. $5.49382$
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  + (0.845 − 0.534i)2-s + (0.278 + 0.960i)3-s + (0.428 − 0.903i)4-s + (−0.987 − 0.160i)5-s + (0.748 + 0.663i)6-s + (−0.120 − 0.992i)8-s + (−0.845 + 0.534i)9-s + (−0.919 + 0.391i)10-s + (0.845 + 0.534i)11-s + (0.987 + 0.160i)12-s + (−0.120 − 0.992i)15-s + (−0.632 − 0.774i)16-s + (0.799 − 0.600i)17-s + (−0.428 + 0.903i)18-s + (0.5 − 0.866i)19-s + (−0.568 + 0.822i)20-s + ⋯
L(s)  = 1  + (0.845 − 0.534i)2-s + (0.278 + 0.960i)3-s + (0.428 − 0.903i)4-s + (−0.987 − 0.160i)5-s + (0.748 + 0.663i)6-s + (−0.120 − 0.992i)8-s + (−0.845 + 0.534i)9-s + (−0.919 + 0.391i)10-s + (0.845 + 0.534i)11-s + (0.987 + 0.160i)12-s + (−0.120 − 0.992i)15-s + (−0.632 − 0.774i)16-s + (0.799 − 0.600i)17-s + (−0.428 + 0.903i)18-s + (0.5 − 0.866i)19-s + (−0.568 + 0.822i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.915 - 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.915 - 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.915 - 0.401i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (298, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ 0.915 - 0.401i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.363655443 - 0.4957376267i\)
\(L(\frac12)\) \(\approx\) \(2.363655443 - 0.4957376267i\)
\(L(1)\) \(\approx\) \(1.643711965 - 0.2012285444i\)
\(L(1)\) \(\approx\) \(1.643711965 - 0.2012285444i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.845 - 0.534i)T \)
3 \( 1 + (0.278 + 0.960i)T \)
5 \( 1 + (-0.987 - 0.160i)T \)
11 \( 1 + (0.845 + 0.534i)T \)
17 \( 1 + (0.799 - 0.600i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.885 + 0.464i)T \)
31 \( 1 + (0.200 - 0.979i)T \)
37 \( 1 + (-0.948 + 0.316i)T \)
41 \( 1 + (0.970 + 0.239i)T \)
43 \( 1 + (-0.748 + 0.663i)T \)
47 \( 1 + (-0.428 - 0.903i)T \)
53 \( 1 + (0.799 - 0.600i)T \)
59 \( 1 + (0.632 - 0.774i)T \)
61 \( 1 + (0.799 + 0.600i)T \)
67 \( 1 + (0.996 + 0.0804i)T \)
71 \( 1 + (0.970 + 0.239i)T \)
73 \( 1 + (0.845 + 0.534i)T \)
79 \( 1 + (0.428 + 0.903i)T \)
83 \( 1 + (0.970 - 0.239i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (0.354 - 0.935i)T \)
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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

−21.30574341396169651562573979346, −20.4875326506591773740075842809, −19.66886947701682449400267229262, −19.126129323460477135173783287455, −18.22568146217661338071834981528, −17.29627081833501896588180140058, −16.50516777926446968035722115249, −15.83506904998494960858200427533, −14.771011407715419674580382668647, −14.30189590828391932130152169840, −13.71235998187237376797693319403, −12.41644411497980108371546873479, −12.26162696239320251446638726716, −11.51690283265750236770760508924, −10.4889101958716483927453905560, −8.88481783570855228780528602061, −8.19227407215343266407620578155, −7.665577222609373438347060541171, −6.697188989420209884348032874, −6.18814337244719152127133929972, −5.125143553839436684319433473129, −3.82042643554830066494803590617, −3.45576312638420727009676806599, −2.34311313125271478506071190368, −1.03652155959182095054279585688, 0.90384602806733042376002644528, 2.328720135649609453624542893911, 3.386244638437812600576177399756, 3.837433665905786761102464991805, 4.78923261249732235591696218149, 5.29600487574455960784125470027, 6.60699292280747538584445739620, 7.51317532956285113476850500616, 8.59982942990331530861403405057, 9.597632967505510589306227214901, 10.06422739811994636845913200752, 11.35959254784305043176668225990, 11.54992021924357790417946109365, 12.40885992898740796828701661216, 13.49548717052576208664900138396, 14.30379200247428566736773709876, 14.90336140071828787425621739982, 15.670956927992869439066001428982, 16.134999651821029611543269203725, 17.09001467609168939583291995821, 18.297121752433617166348110686482, 19.46444227441273928903166323584, 19.74284085465895304906059332763, 20.42473259770671025389196684027, 21.16168998107984232525521253523

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