Properties

Label 2-17e2-17.13-c1-0-11
Degree $2$
Conductor $289$
Sign $-0.676 + 0.736i$
Analytic cond. $2.30767$
Root an. cond. $1.51910$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.41i·2-s + (0.765 + 0.765i)3-s − 3.82·4-s + (0.541 + 0.541i)5-s + (1.84 − 1.84i)6-s + (1.84 − 1.84i)7-s + 4.41i·8-s − 1.82i·9-s + (1.30 − 1.30i)10-s + (1.84 − 1.84i)11-s + (−2.93 − 2.93i)12-s − 1.41·13-s + (−4.46 − 4.46i)14-s + 0.828i·15-s + 2.99·16-s + ⋯
L(s)  = 1  − 1.70i·2-s + (0.441 + 0.441i)3-s − 1.91·4-s + (0.242 + 0.242i)5-s + (0.754 − 0.754i)6-s + (0.698 − 0.698i)7-s + 1.56i·8-s − 0.609i·9-s + (0.413 − 0.413i)10-s + (0.557 − 0.557i)11-s + (−0.845 − 0.845i)12-s − 0.392·13-s + (−1.19 − 1.19i)14-s + 0.213i·15-s + 0.749·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.676 + 0.736i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.676 + 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $-0.676 + 0.736i$
Analytic conductor: \(2.30767\)
Root analytic conductor: \(1.51910\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{289} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 289,\ (\ :1/2),\ -0.676 + 0.736i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.584109 - 1.33057i\)
\(L(\frac12)\) \(\approx\) \(0.584109 - 1.33057i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
good2 \( 1 + 2.41iT - 2T^{2} \)
3 \( 1 + (-0.765 - 0.765i)T + 3iT^{2} \)
5 \( 1 + (-0.541 - 0.541i)T + 5iT^{2} \)
7 \( 1 + (-1.84 + 1.84i)T - 7iT^{2} \)
11 \( 1 + (-1.84 + 1.84i)T - 11iT^{2} \)
13 \( 1 + 1.41T + 13T^{2} \)
19 \( 1 - 0.828iT - 19T^{2} \)
23 \( 1 + (3.37 - 3.37i)T - 23iT^{2} \)
29 \( 1 + (0.224 + 0.224i)T + 29iT^{2} \)
31 \( 1 + (-5.54 - 5.54i)T + 31iT^{2} \)
37 \( 1 + (-6.53 - 6.53i)T + 37iT^{2} \)
41 \( 1 + (0.858 - 0.858i)T - 41iT^{2} \)
43 \( 1 - 0.828iT - 43T^{2} \)
47 \( 1 + 5.17T + 47T^{2} \)
53 \( 1 + 1.41iT - 53T^{2} \)
59 \( 1 - 6iT - 59T^{2} \)
61 \( 1 + (2.70 - 2.70i)T - 61iT^{2} \)
67 \( 1 + 1.17T + 67T^{2} \)
71 \( 1 + (3.82 + 3.82i)T + 71iT^{2} \)
73 \( 1 + (-9.14 - 9.14i)T + 73iT^{2} \)
79 \( 1 + (3.37 - 3.37i)T - 79iT^{2} \)
83 \( 1 + 11.6iT - 83T^{2} \)
89 \( 1 + 6.58T + 89T^{2} \)
97 \( 1 + (-7.29 - 7.29i)T + 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.54254515347774511389160047794, −10.42136867397079034839702595845, −9.932636080318640029008638285062, −9.007304124580036678072255506403, −8.043656590833277101642777312116, −6.41617942331263629066068175608, −4.67722716815156011943295611576, −3.83046329924954175392518814145, −2.80465036338089220196351953671, −1.22741314662488303593048439230, 2.14545997283557492071783253481, 4.45932276436874365070804502745, 5.30437005799406043053290783493, 6.34213445668351791831691858949, 7.42169080461076367041695851932, 8.079148959178674583818507497684, 8.874772067504507708945711563038, 9.752440429472653758084701928189, 11.32272856776064232585622541501, 12.50348462568591924513558425876

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