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Properties

Label 2-17e2-17.4-c1-0-8
Degree $2$
Conductor $289$
Sign $0.988 + 0.151i$
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  − i·2-s + 4-s + (−1.41 + 1.41i)5-s + (2.82 + 2.82i)7-s − 3i·8-s + 3i·9-s + (1.41 + 1.41i)10-s + 2·13-s + (2.82 − 2.82i)14-s − 16-s + 3·18-s − 4i·19-s + (−1.41 + 1.41i)20-s + (−2.82 − 2.82i)23-s + 0.999i·25-s − 2i·26-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.5·4-s + (−0.632 + 0.632i)5-s + (1.06 + 1.06i)7-s − 1.06i·8-s + i·9-s + (0.447 + 0.447i)10-s + 0.554·13-s + (0.755 − 0.755i)14-s − 0.250·16-s + 0.707·18-s − 0.917i·19-s + (−0.316 + 0.316i)20-s + (−0.589 − 0.589i)23-s + 0.199i·25-s − 0.392i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.151i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 + 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $0.988 + 0.151i$
Analytic conductor: \(2.30767\)
Root analytic conductor: \(1.51910\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{289} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 289,\ (\ :1/2),\ 0.988 + 0.151i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49656 - 0.113896i\)
\(L(\frac12)\) \(\approx\) \(1.49656 - 0.113896i\)
\(L(\frac{3}{2})\) 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 \( 1 + iT - 2T^{2} \)
3 \( 1 - 3iT^{2} \)
5 \( 1 + (1.41 - 1.41i)T - 5iT^{2} \)
7 \( 1 + (-2.82 - 2.82i)T + 7iT^{2} \)
11 \( 1 + 11iT^{2} \)
13 \( 1 - 2T + 13T^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + (2.82 + 2.82i)T + 23iT^{2} \)
29 \( 1 + (-4.24 + 4.24i)T - 29iT^{2} \)
31 \( 1 + (2.82 - 2.82i)T - 31iT^{2} \)
37 \( 1 + (-1.41 + 1.41i)T - 37iT^{2} \)
41 \( 1 + (4.24 + 4.24i)T + 41iT^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 - 12iT - 59T^{2} \)
61 \( 1 + (7.07 + 7.07i)T + 61iT^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + (-2.82 + 2.82i)T - 71iT^{2} \)
73 \( 1 + (4.24 - 4.24i)T - 73iT^{2} \)
79 \( 1 + (8.48 + 8.48i)T + 79iT^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + (-1.41 + 1.41i)T - 97iT^{2} \)
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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

−11.60322312778789218017388134955, −11.01662592075909478669293824960, −10.36556613432651476977829617567, −8.883585088393112101613262992568, −7.952949940978610462789718065444, −7.02384080320237231949608930726, −5.72340541360265930522700504218, −4.41851322054518861913861654927, −2.92296580188787187385172806572, −1.93173898681769504718790134616, 1.37514941315101394633710094793, 3.63971252735354225242146111423, 4.71505860949889972998330787989, 5.99082418099572396806069248176, 7.03562764123495149111898619858, 7.982611252170780811741778749805, 8.486157206614827493145003420986, 9.978041570622829359021031828535, 11.11253574653111584618740140806, 11.71093293024900999763109868049

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