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Properties

Label 2-650-65.32-c1-0-20
Degree $2$
Conductor $650$
Sign $-0.173 - 0.984i$
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  + (0.5 − 0.866i)2-s + (−0.633 − 2.36i)3-s + (−0.499 − 0.866i)4-s + (−2.36 − 0.633i)6-s + (−3.23 + 1.86i)7-s − 0.999·8-s + (−2.59 + 1.50i)9-s + (1.86 − 0.5i)11-s + (−1.73 + 1.73i)12-s + (−3.5 + 0.866i)13-s + 3.73i·14-s + (−0.5 + 0.866i)16-s + (−5.09 − 1.36i)17-s + 3.00i·18-s + (−1.33 + 4.96i)19-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.366 − 1.36i)3-s + (−0.249 − 0.433i)4-s + (−0.965 − 0.258i)6-s + (−1.22 + 0.705i)7-s − 0.353·8-s + (−0.866 + 0.500i)9-s + (0.562 − 0.150i)11-s + (−0.499 + 0.499i)12-s + (−0.970 + 0.240i)13-s + 0.997i·14-s + (−0.125 + 0.216i)16-s + (−1.23 − 0.331i)17-s + 0.707i·18-s + (−0.305 + 1.13i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $-0.173 - 0.984i$
Analytic conductor: \(5.19027\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{650} (357, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 650,\ (\ :1/2),\ -0.173 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.189797 + 0.226063i\)
\(L(\frac12)\) \(\approx\) \(0.189797 + 0.226063i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 \)
13 \( 1 + (3.5 - 0.866i)T \)
good3 \( 1 + (0.633 + 2.36i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (3.23 - 1.86i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.86 + 0.5i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (5.09 + 1.36i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.33 - 4.96i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-3.36 + 0.901i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (8.19 + 4.73i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3 + 3i)T - 31iT^{2} \)
37 \( 1 + (-3.23 - 1.86i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.0980 + 0.366i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (2.19 - 8.19i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + 7iT - 47T^{2} \)
53 \( 1 + (-6.36 + 6.36i)T - 53iT^{2} \)
59 \( 1 + (-0.633 - 0.169i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (6.36 + 11.0i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.464 - 0.803i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (10.4 + 2.80i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + 4.73T + 73T^{2} \)
79 \( 1 - 11.1iT - 79T^{2} \)
83 \( 1 + 2.73iT - 83T^{2} \)
89 \( 1 + (-3.89 - 14.5i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (8.09 + 14.0i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.821371069715043246497199750347, −9.289855390104693625846348161778, −8.124344189379708629369057515666, −6.93832051087388029976092590495, −6.37997635043578352329513323815, −5.58197031356874866926470099310, −4.16094435278798708542176292574, −2.78417689451588635895496511497, −1.87121464826895436145828200950, −0.13903323691364900510645126765, 2.95680687394568171862727136436, 4.02252305833928134487643461747, 4.63315903179647416607223536445, 5.65395754210595225026923864539, 6.75715534573429797310438413960, 7.29831007788107815896941639390, 9.058293977185890415093201714010, 9.251948294865530786685090989090, 10.34371864524502037984765103710, 10.88846828529024265237776998523

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