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Properties

Label 1-13e2-169.4-r0-0-0
Degree $1$
Conductor $169$
Sign $0.858 + 0.513i$
Analytic cond. $0.784832$
Root an. cond. $0.784832$
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  + (0.996 + 0.0804i)2-s + (−0.845 − 0.534i)3-s + (0.987 + 0.160i)4-s + (0.748 + 0.663i)5-s + (−0.799 − 0.600i)6-s + (−0.692 + 0.721i)7-s + (0.970 + 0.239i)8-s + (0.428 + 0.903i)9-s + (0.692 + 0.721i)10-s + (−0.428 + 0.903i)11-s + (−0.748 − 0.663i)12-s + (−0.748 + 0.663i)14-s + (−0.278 − 0.960i)15-s + (0.948 + 0.316i)16-s + (0.692 − 0.721i)17-s + (0.354 + 0.935i)18-s + ⋯
L(s)  = 1  + (0.996 + 0.0804i)2-s + (−0.845 − 0.534i)3-s + (0.987 + 0.160i)4-s + (0.748 + 0.663i)5-s + (−0.799 − 0.600i)6-s + (−0.692 + 0.721i)7-s + (0.970 + 0.239i)8-s + (0.428 + 0.903i)9-s + (0.692 + 0.721i)10-s + (−0.428 + 0.903i)11-s + (−0.748 − 0.663i)12-s + (−0.748 + 0.663i)14-s + (−0.278 − 0.960i)15-s + (0.948 + 0.316i)16-s + (0.692 − 0.721i)17-s + (0.354 + 0.935i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.858 + 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.858 + 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.858 + 0.513i$
Analytic conductor: \(0.784832\)
Root analytic conductor: \(0.784832\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 169,\ (0:\ ),\ 0.858 + 0.513i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.636180567 + 0.4520247803i\)
\(L(\frac12)\) \(\approx\) \(1.636180567 + 0.4520247803i\)
\(L(1)\) \(\approx\) \(1.527997604 + 0.2072552678i\)
\(L(1)\) \(\approx\) \(1.527997604 + 0.2072552678i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
good2 \( 1 + (0.996 + 0.0804i)T \)
3 \( 1 + (-0.845 - 0.534i)T \)
5 \( 1 + (0.748 + 0.663i)T \)
7 \( 1 + (-0.692 + 0.721i)T \)
11 \( 1 + (-0.428 + 0.903i)T \)
17 \( 1 + (0.692 - 0.721i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.996 - 0.0804i)T \)
31 \( 1 + (-0.120 + 0.992i)T \)
37 \( 1 + (0.919 - 0.391i)T \)
41 \( 1 + (0.845 + 0.534i)T \)
43 \( 1 + (-0.919 - 0.391i)T \)
47 \( 1 + (0.354 - 0.935i)T \)
53 \( 1 + (-0.970 - 0.239i)T \)
59 \( 1 + (-0.948 + 0.316i)T \)
61 \( 1 + (0.278 - 0.960i)T \)
67 \( 1 + (-0.987 + 0.160i)T \)
71 \( 1 + (0.0402 - 0.999i)T \)
73 \( 1 + (-0.568 - 0.822i)T \)
79 \( 1 + (-0.354 + 0.935i)T \)
83 \( 1 + (-0.885 - 0.464i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (0.200 + 0.979i)T \)
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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

−27.80399014858963081410956559563, −26.404068975725645585186999340068, −25.49944529511155980515926338187, −24.18025781278137609484787076864, −23.6210585192830541304719474978, −22.576739764800072016823765103, −21.75400428482092193838078772193, −20.98311263342123746198418206542, −20.172152777022018668562910344824, −18.76278615641215213479811630233, −17.16075729155878825894227161940, −16.52782274561864525674978959470, −15.84336993567711076287666715286, −14.42168299029528343270917604733, −13.3328233107031038581993736319, −12.634877724424622885955760212805, −11.44596521217674182527726713666, −10.36840512320603598152259761965, −9.61316685190123333284470781285, −7.64821285460505199412528670726, −5.975856618989131013545027486829, −5.77018727568231058423985043065, −4.32878531164366477487955309188, −3.321677714282282612612168681255, −1.2950985933048639545071501829, 1.988537996902503939727866542480, 2.96567679505866990521134502760, 4.88960612921600251924267685963, 5.760655481839085896978163294987, 6.67098197406955179539599549305, 7.50121448328498374343105292739, 9.66741961135997102750629090388, 10.70627806701887212779000898474, 11.83433699355330429606776848838, 12.69707406913726122860092673594, 13.48893254735573384988595341905, 14.64436890773459302585214384650, 15.74872450756426151536205728256, 16.680601596408548439988332559973, 17.94069427792668060415979713063, 18.65637047422521941397139595091, 19.98772297964632793848617620415, 21.306603044358735022341357798208, 22.16853163803668960703494988, 22.70776443104092436924592356811, 23.55838150217176059388070681395, 24.82458154971725182755210337923, 25.3410234802720407707098966407, 26.36349866925423427512276172969, 28.30559323653021280004073722479

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