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Properties

Label 1-1183-1183.748-r0-0-0
Degree $1$
Conductor $1183$
Sign $0.931 - 0.363i$
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  + (−0.774 − 0.632i)2-s + (0.996 + 0.0804i)3-s + (0.200 + 0.979i)4-s + (0.992 − 0.120i)5-s + (−0.721 − 0.692i)6-s + (0.464 − 0.885i)8-s + (0.987 + 0.160i)9-s + (−0.845 − 0.534i)10-s + (−0.160 − 0.987i)11-s + (0.120 + 0.992i)12-s + (0.999 − 0.0402i)15-s + (−0.919 + 0.391i)16-s + (−0.845 + 0.534i)17-s + (−0.663 − 0.748i)18-s + (0.866 + 0.5i)19-s + (0.316 + 0.948i)20-s + ⋯
L(s)  = 1  + (−0.774 − 0.632i)2-s + (0.996 + 0.0804i)3-s + (0.200 + 0.979i)4-s + (0.992 − 0.120i)5-s + (−0.721 − 0.692i)6-s + (0.464 − 0.885i)8-s + (0.987 + 0.160i)9-s + (−0.845 − 0.534i)10-s + (−0.160 − 0.987i)11-s + (0.120 + 0.992i)12-s + (0.999 − 0.0402i)15-s + (−0.919 + 0.391i)16-s + (−0.845 + 0.534i)17-s + (−0.663 − 0.748i)18-s + (0.866 + 0.5i)19-s + (0.316 + 0.948i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.931 - 0.363i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.931 - 0.363i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1183\)    =    \(7 \cdot 13^{2}\)
Sign: $0.931 - 0.363i$
Analytic conductor: \(5.49382\)
Root analytic conductor: \(5.49382\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1183} (748, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1183,\ (0:\ ),\ 0.931 - 0.363i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.889300323 - 0.3553099958i\)
\(L(\frac12)\) \(\approx\) \(1.889300323 - 0.3553099958i\)
\(L(1)\) \(\approx\) \(1.258621966 - 0.2437118477i\)
\(L(1)\) \(\approx\) \(1.258621966 - 0.2437118477i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.774 - 0.632i)T \)
3 \( 1 + (0.996 + 0.0804i)T \)
5 \( 1 + (0.992 - 0.120i)T \)
11 \( 1 + (-0.160 - 0.987i)T \)
17 \( 1 + (-0.845 + 0.534i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.632 + 0.774i)T \)
31 \( 1 + (-0.239 + 0.970i)T \)
37 \( 1 + (0.960 - 0.278i)T \)
41 \( 1 + (-0.0804 + 0.996i)T \)
43 \( 1 + (-0.278 + 0.960i)T \)
47 \( 1 + (0.663 - 0.748i)T \)
53 \( 1 + (0.885 + 0.464i)T \)
59 \( 1 + (0.391 - 0.919i)T \)
61 \( 1 + (0.0402 - 0.999i)T \)
67 \( 1 + (-0.979 - 0.200i)T \)
71 \( 1 + (-0.903 + 0.428i)T \)
73 \( 1 + (-0.935 - 0.354i)T \)
79 \( 1 + (-0.748 - 0.663i)T \)
83 \( 1 + (0.822 - 0.568i)T \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + (0.600 - 0.799i)T \)
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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

−20.67534538381420746602260969151, −20.62289611321812357501474188018, −19.70445649510693340246450341155, −18.738660907238261365125973884748, −18.202758142832809893578584397070, −17.57423330549205845267009844640, −16.72653071268427819879340944191, −15.75495994180522892716951804231, −15.043890571725388730381476804505, −14.51140373481112498494548994056, −13.51571621492724420272004256528, −13.17451731254272843455050157639, −11.78135982570578198085486464831, −10.584434674797757728447754927124, −9.943039730614499825020274193444, −9.208654633886952857931934325606, −8.79714260921336713399357817953, −7.5100740390429299780377860976, −7.139609525151697697108038227538, −6.18504110901514505806231049619, −5.14237734052496945296637946551, −4.25469207607634040551982341906, −2.54568580770720831176118869140, −2.227665786488229487973916167968, −1.02528471244031766335075412098, 1.20969094298558454059029366956, 1.87898574287897487115129533149, 2.94303318154059760949951110816, 3.480289335963798328791957843275, 4.70914062664210532507465486423, 5.926778958028779012346747324882, 7.00350615967526920088251995523, 7.85593185717589183059352346196, 8.78267546871121822385827890724, 9.15904487842702854745866897738, 10.02313375976960305426360704277, 10.699925954275853215862246482694, 11.58263065929346111986848032727, 12.86536188603201398700042815455, 13.23160669259914378018133614594, 13.99923252424376149528558127955, 14.90523342201229501202343586896, 16.0424151978923246319685344971, 16.537929192241137224348856705430, 17.57526376920320171069123989874, 18.26584219268727601742666613446, 18.85264394996339090252090248886, 19.79590124757947573965696086392, 20.24106509114631755469992487934, 21.12691848741974855450069879700

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