Properties

Label 1-13e2-169.10-r0-0-0
Degree $1$
Conductor $169$
Sign $0.999 + 0.0371i$
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  + (0.919 + 0.391i)2-s + (0.948 − 0.316i)3-s + (0.692 + 0.721i)4-s + (−0.885 − 0.464i)5-s + (0.996 + 0.0804i)6-s + (0.632 − 0.774i)7-s + (0.354 + 0.935i)8-s + (0.799 − 0.600i)9-s + (−0.632 − 0.774i)10-s + (−0.799 − 0.600i)11-s + (0.885 + 0.464i)12-s + (0.885 − 0.464i)14-s + (−0.987 − 0.160i)15-s + (−0.0402 + 0.999i)16-s + (−0.632 + 0.774i)17-s + (0.970 − 0.239i)18-s + ⋯
L(s)  = 1  + (0.919 + 0.391i)2-s + (0.948 − 0.316i)3-s + (0.692 + 0.721i)4-s + (−0.885 − 0.464i)5-s + (0.996 + 0.0804i)6-s + (0.632 − 0.774i)7-s + (0.354 + 0.935i)8-s + (0.799 − 0.600i)9-s + (−0.632 − 0.774i)10-s + (−0.799 − 0.600i)11-s + (0.885 + 0.464i)12-s + (0.885 − 0.464i)14-s + (−0.987 − 0.160i)15-s + (−0.0402 + 0.999i)16-s + (−0.632 + 0.774i)17-s + (0.970 − 0.239i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0371i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0371i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.999 + 0.0371i$
Analytic conductor: \(0.784832\)
Root analytic conductor: \(0.784832\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 169,\ (0:\ ),\ 0.999 + 0.0371i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.280697101 + 0.04240145972i\)
\(L(\frac12)\) \(\approx\) \(2.280697101 + 0.04240145972i\)
\(L(1)\) \(\approx\) \(2.000107316 + 0.08601707960i\)
\(L(1)\) \(\approx\) \(2.000107316 + 0.08601707960i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
good2 \( 1 + (0.919 + 0.391i)T \)
3 \( 1 + (0.948 - 0.316i)T \)
5 \( 1 + (-0.885 - 0.464i)T \)
7 \( 1 + (0.632 - 0.774i)T \)
11 \( 1 + (-0.799 - 0.600i)T \)
17 \( 1 + (-0.632 + 0.774i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.919 - 0.391i)T \)
31 \( 1 + (-0.568 + 0.822i)T \)
37 \( 1 + (-0.428 - 0.903i)T \)
41 \( 1 + (-0.948 + 0.316i)T \)
43 \( 1 + (0.428 - 0.903i)T \)
47 \( 1 + (0.970 + 0.239i)T \)
53 \( 1 + (-0.354 - 0.935i)T \)
59 \( 1 + (0.0402 + 0.999i)T \)
61 \( 1 + (0.987 - 0.160i)T \)
67 \( 1 + (-0.692 + 0.721i)T \)
71 \( 1 + (0.200 - 0.979i)T \)
73 \( 1 + (-0.120 + 0.992i)T \)
79 \( 1 + (-0.970 - 0.239i)T \)
83 \( 1 + (0.748 - 0.663i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (0.845 + 0.534i)T \)
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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

−27.69704815121646168471563907012, −26.5760894396967518288699186755, −25.589804982333377480010469173157, −24.44543943976867546870823581821, −23.84312879977891065931383360072, −22.445309520993118257837608860117, −21.89096564401666798644728237209, −20.48266207801020795610722751320, −20.30506218822801128580171550733, −18.92632018558751000253489886542, −18.29421998555044038450846549583, −16.03027062820888294677533812134, −15.35120505685416799380295094721, −14.77243028998316358448011124139, −13.71048137307938039477937182457, −12.595458751428650607487855849954, −11.50682408543576369796269962582, −10.59589440068110708137008786882, −9.28521239788963812962386202674, −7.92779599724227867653464776854, −6.974498681555997683336212052581, −5.13504474647716998613256633705, −4.29073526443356523599640849553, −2.95887408946182497827941647487, −2.17322720688558272120270513185, 1.77046881363919153279188190979, 3.46030611985891959523807550382, 4.11329593576971536436018405269, 5.48855145812160825365494352611, 7.19632280122274947147950331331, 7.85319425204312663123008777955, 8.65179697312454124878329160751, 10.600679449243938608296796226598, 11.76394607049975877571293369254, 12.87203144036462820673712794969, 13.64329408294133710277176521942, 14.58732371918350905163301241464, 15.5347777342142562258141036139, 16.35503258385351227008718083864, 17.64778643443541914161383208665, 19.07952813642984488076490708474, 20.13794274495269170470611084005, 20.67494421632537062230658124931, 21.64474984422732124567968410111, 23.13536141760137758836993081633, 24.002991671512328610476639684129, 24.26802943305318884246194136103, 25.53128179506949034120134797453, 26.56121136785954066283784655913, 27.12721165799660178132674713171

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