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Properties

Label 2-650-65.18-c1-0-1
Degree $2$
Conductor $650$
Sign $0.966 - 0.256i$
Analytic cond. $5.19027$
Root an. cond. $2.27821$
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 + (−2.15 − 2.15i)3-s − 4-s + (−2.15 + 2.15i)6-s − 3·7-s + i·8-s + 6.31i·9-s + (−3.31 − 3.31i)11-s + (2.15 + 2.15i)12-s + (3 + 2i)13-s + 3i·14-s + 16-s + (−0.158 − 0.158i)17-s + 6.31·18-s + (2 + 2i)19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−1.24 − 1.24i)3-s − 0.5·4-s + (−0.881 + 0.881i)6-s − 1.13·7-s + 0.353i·8-s + 2.10i·9-s + (−1.00 − 1.00i)11-s + (0.623 + 0.623i)12-s + (0.832 + 0.554i)13-s + 0.801i·14-s + 0.250·16-s + (−0.0383 − 0.0383i)17-s + 1.48·18-s + (0.458 + 0.458i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $0.966 - 0.256i$
Analytic conductor: \(5.19027\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :1/2),\ 0.966 - 0.256i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.258984 + 0.0338026i\)
\(L(\frac12)\) \(\approx\) \(0.258984 + 0.0338026i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 \)
13 \( 1 + (-3 - 2i)T \)
good3 \( 1 + (2.15 + 2.15i)T + 3iT^{2} \)
7 \( 1 + 3T + 7T^{2} \)
11 \( 1 + (3.31 + 3.31i)T + 11iT^{2} \)
17 \( 1 + (0.158 + 0.158i)T + 17iT^{2} \)
19 \( 1 + (-2 - 2i)T + 19iT^{2} \)
23 \( 1 + (3.31 - 3.31i)T - 23iT^{2} \)
29 \( 1 + 0.316iT - 29T^{2} \)
31 \( 1 + (-1 + i)T - 31iT^{2} \)
37 \( 1 + 3T + 37T^{2} \)
41 \( 1 + (-0.316 + 0.316i)T - 41iT^{2} \)
43 \( 1 + (-7.47 + 7.47i)T - 43iT^{2} \)
47 \( 1 - 2.68T + 47T^{2} \)
53 \( 1 + (-3.63 - 3.63i)T + 53iT^{2} \)
59 \( 1 + (6.31 - 6.31i)T - 59iT^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 14.9iT - 67T^{2} \)
71 \( 1 + (6.15 - 6.15i)T - 71iT^{2} \)
73 \( 1 - 4iT - 73T^{2} \)
79 \( 1 - 6.94iT - 79T^{2} \)
83 \( 1 - 0.316T + 83T^{2} \)
89 \( 1 + (6.31 - 6.31i)T - 89iT^{2} \)
97 \( 1 + 10.9iT - 97T^{2} \)
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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

−10.78608884095923555286312630360, −10.08228078010265413158317353150, −8.872797717130488290677626748784, −7.85114919670891280299127499543, −6.92433097535293963292104063782, −5.89813783862023931985044430609, −5.54506486243061686423966593083, −3.86462063301999775614997396571, −2.57639274777543183794242572321, −1.12158650166937016435379320013, 0.19368768241543574856012226815, 3.18004060235087151523835304498, 4.30082328009033471300480176205, 5.11266964402123890919698020234, 5.97085090607775516639289266528, 6.60781118541089183983714097767, 7.76483805899297988098998180419, 9.044609489427900747140167875845, 9.814275537442040830643126475904, 10.35232917002456124354936705563

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