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Properties

Label 1-13e2-169.12-r0-0-0
Degree $1$
Conductor $169$
Sign $0.656 + 0.754i$
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  + (0.354 − 0.935i)2-s + (0.568 + 0.822i)3-s + (−0.748 − 0.663i)4-s + (−0.120 + 0.992i)5-s + (0.970 − 0.239i)6-s + (−0.885 + 0.464i)7-s + (−0.885 + 0.464i)8-s + (−0.354 + 0.935i)9-s + (0.885 + 0.464i)10-s + (0.354 + 0.935i)11-s + (0.120 − 0.992i)12-s + (0.120 + 0.992i)14-s + (−0.885 + 0.464i)15-s + (0.120 + 0.992i)16-s + (0.885 − 0.464i)17-s + (0.748 + 0.663i)18-s + ⋯
L(s)  = 1  + (0.354 − 0.935i)2-s + (0.568 + 0.822i)3-s + (−0.748 − 0.663i)4-s + (−0.120 + 0.992i)5-s + (0.970 − 0.239i)6-s + (−0.885 + 0.464i)7-s + (−0.885 + 0.464i)8-s + (−0.354 + 0.935i)9-s + (0.885 + 0.464i)10-s + (0.354 + 0.935i)11-s + (0.120 − 0.992i)12-s + (0.120 + 0.992i)14-s + (−0.885 + 0.464i)15-s + (0.120 + 0.992i)16-s + (0.885 − 0.464i)17-s + (0.748 + 0.663i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.656 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.656 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.656 + 0.754i$
Analytic conductor: \(0.784832\)
Root analytic conductor: \(0.784832\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 169,\ (0:\ ),\ 0.656 + 0.754i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.104777976 + 0.5034074367i\)
\(L(\frac12)\) \(\approx\) \(1.104777976 + 0.5034074367i\)
\(L(1)\) \(\approx\) \(1.173533063 + 0.1188984076i\)
\(L(1)\) \(\approx\) \(1.173533063 + 0.1188984076i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
good2 \( 1 + (0.354 - 0.935i)T \)
3 \( 1 + (0.568 + 0.822i)T \)
5 \( 1 + (-0.120 + 0.992i)T \)
7 \( 1 + (-0.885 + 0.464i)T \)
11 \( 1 + (0.354 + 0.935i)T \)
17 \( 1 + (0.885 - 0.464i)T \)
19 \( 1 - T \)
23 \( 1 + T \)
29 \( 1 + (-0.354 + 0.935i)T \)
31 \( 1 + (0.970 - 0.239i)T \)
37 \( 1 + (0.970 - 0.239i)T \)
41 \( 1 + (-0.568 - 0.822i)T \)
43 \( 1 + (-0.970 - 0.239i)T \)
47 \( 1 + (0.748 - 0.663i)T \)
53 \( 1 + (0.885 - 0.464i)T \)
59 \( 1 + (-0.120 + 0.992i)T \)
61 \( 1 + (0.885 + 0.464i)T \)
67 \( 1 + (0.748 - 0.663i)T \)
71 \( 1 + (-0.568 - 0.822i)T \)
73 \( 1 + (0.354 + 0.935i)T \)
79 \( 1 + (-0.748 + 0.663i)T \)
83 \( 1 + (-0.568 + 0.822i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.120 - 0.992i)T \)
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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

−27.10985892255254258594565384212, −26.23921938242376943863022032461, −25.1950350119481762235101102942, −24.75509680485978008180818296186, −23.54804507740650919164022639524, −23.257895643057651352732190063401, −21.67976783007577297412526485016, −20.68565722711301505439864084397, −19.415898043115133800668402647943, −18.80277450324833091325002207398, −17.14698103647915285569137854238, −16.77841386536151134077251908106, −15.55225951465972823072755828294, −14.42556164508728938408831435822, −13.31107484687123055266802127570, −12.938615882709883571768722298564, −11.82481177856335109249342399406, −9.67466402615091589866249375569, −8.648316892058704431471223857386, −7.92975994981092918564345269010, −6.65539861397261889131629379250, −5.82165281477140183653201183004, −4.19106485432020472178897601787, −3.12584966775504834103670446799, −0.87142293889085359309468511087, 2.28364497545029478106107681524, 3.14993662740459932858167959420, 4.11559400731454537962159302133, 5.480707160035298563225969667, 6.94498081755422707423010135494, 8.69281111433198665734361967063, 9.757650563032016742015172383721, 10.32682633560728548994956543822, 11.49000664026208161572484191128, 12.63642021490809190671843414262, 13.80735724352431865063086943256, 14.878616095206497619887603636224, 15.29216971277773726321092931134, 16.8536165540860006234284913506, 18.38306988548941576981180337713, 19.175509307699671639148743073010, 19.91960716272246769929602633985, 21.023120347603197813289624919914, 21.86393332316879774885877718659, 22.65578147991068820035560462027, 23.24958308466661218911591554374, 25.20516328729682157598266430013, 25.85212905215967989683609788672, 26.973708489422545726532621422706, 27.70606718959656006472473718731

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