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Properties

Label 1-1649-1649.1161-r1-0-0
Degree $1$
Conductor $1649$
Sign $0.801 - 0.597i$
Analytic cond. $177.209$
Root an. cond. $177.209$
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  + (0.5 + 0.866i)2-s + (−0.608 + 0.793i)3-s + (−0.5 + 0.866i)4-s + (0.258 − 0.965i)5-s + (−0.991 − 0.130i)6-s + (−0.965 − 0.258i)7-s − 8-s + (−0.258 − 0.965i)9-s + (0.965 − 0.258i)10-s + (0.793 − 0.608i)11-s + (−0.382 − 0.923i)12-s + (0.991 − 0.130i)13-s + (−0.258 − 0.965i)14-s + (0.608 + 0.793i)15-s + (−0.5 − 0.866i)16-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.608 + 0.793i)3-s + (−0.5 + 0.866i)4-s + (0.258 − 0.965i)5-s + (−0.991 − 0.130i)6-s + (−0.965 − 0.258i)7-s − 8-s + (−0.258 − 0.965i)9-s + (0.965 − 0.258i)10-s + (0.793 − 0.608i)11-s + (−0.382 − 0.923i)12-s + (0.991 − 0.130i)13-s + (−0.258 − 0.965i)14-s + (0.608 + 0.793i)15-s + (−0.5 − 0.866i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1649 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.801 - 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1649 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.801 - 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1649\)    =    \(17 \cdot 97\)
Sign: $0.801 - 0.597i$
Analytic conductor: \(177.209\)
Root analytic conductor: \(177.209\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1649} (1161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1649,\ (1:\ ),\ 0.801 - 0.597i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.166328232 - 0.3867179419i\)
\(L(\frac12)\) \(\approx\) \(1.166328232 - 0.3867179419i\)
\(L(1)\) \(\approx\) \(0.9079304111 + 0.4000214265i\)
\(L(1)\) \(\approx\) \(0.9079304111 + 0.4000214265i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
97 \( 1 \)
good2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (-0.608 + 0.793i)T \)
5 \( 1 + (0.258 - 0.965i)T \)
7 \( 1 + (-0.965 - 0.258i)T \)
11 \( 1 + (0.793 - 0.608i)T \)
13 \( 1 + (0.991 - 0.130i)T \)
19 \( 1 + (0.923 - 0.382i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.965 + 0.258i)T \)
31 \( 1 + (-0.608 + 0.793i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (0.866 - 0.5i)T \)
43 \( 1 + (-0.965 - 0.258i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (-0.991 + 0.130i)T \)
61 \( 1 + (-0.793 - 0.608i)T \)
67 \( 1 + (-0.923 - 0.382i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + (0.991 - 0.130i)T \)
79 \( 1 + (0.923 + 0.382i)T \)
83 \( 1 + (0.130 - 0.991i)T \)
89 \( 1 + (0.707 + 0.707i)T \)
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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

−20.00816384901190674280458302131, −19.632406240737032879876957954263, −18.66826519294849728229621930858, −18.350338655353756280110670144770, −17.74326947569780737784002867290, −16.61052332398826242515911704025, −15.793344216293859417193501083440, −14.77248616043521451338211695697, −14.056368584158484623543870782286, −13.44761183246752924276865262367, −12.67489514889015447237748584731, −12.008539915621376906007739663740, −11.357133197850270589799061750512, −10.612243710767779802518435789649, −9.85033628232252053472875107083, −9.15223631802078491724081341077, −7.87004677099903092605594333214, −6.75784624444000511804574159724, −6.22841257278392386893773215494, −5.76116289671927920443662687557, −4.47236802954660699189812378895, −3.52851347505784574879754869212, −2.70177919379114254184645601594, −1.87697120814642891273928119510, −0.94854057655369138606899813747, 0.25602040629725806753202350560, 1.194117729915493804564124637134, 3.3234731806842431333958751124, 3.57839852758973056367849820533, 4.64525420626686854502384440831, 5.307792329267184549725103775672, 6.21049882970893084189174599135, 6.48763145747951120667603912885, 7.81404664622619883368419303587, 8.84832773461128155541638079430, 9.25579905745472030331092276120, 10.02868589570073561836359836447, 11.22510830117626860242358493605, 11.99730443726146587584281122261, 12.66382313560382056455711652583, 13.618443823671648115275497559735, 13.96468259032228787950134167824, 15.21561733927584829563650787636, 15.955924190533343720381898400366, 16.28039377541686746472209965335, 16.8297242441593480333281082890, 17.64329597578227769569115549145, 18.24950085460795382988179956418, 19.64757492682164860589436640327, 20.25883826163291800201864175245

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