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Properties

Label 2-350-5.4-c1-0-3
Degree $2$
Conductor $350$
Sign $0.894 - 0.447i$
Analytic cond. $2.79476$
Root an. cond. $1.67175$
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 − i·3-s − 4-s + 6-s + i·7-s − i·8-s + 2·9-s + 3·11-s + i·12-s − 2i·13-s − 14-s + 16-s + 3i·17-s + 2i·18-s + 7·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s + 0.377i·7-s − 0.353i·8-s + 0.666·9-s + 0.904·11-s + 0.288i·12-s − 0.554i·13-s − 0.267·14-s + 0.250·16-s + 0.727i·17-s + 0.471i·18-s + 1.60·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36870 + 0.323107i\)
\(L(\frac12)\) \(\approx\) \(1.36870 + 0.323107i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 - iT \)
good3 \( 1 + iT - 3T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 - 3iT - 17T^{2} \)
19 \( 1 - 7T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 - 8iT - 37T^{2} \)
41 \( 1 + 9T + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 + 6iT - 47T^{2} \)
53 \( 1 - 12iT - 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + 7iT - 67T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + 5iT - 73T^{2} \)
79 \( 1 + 14T + 79T^{2} \)
83 \( 1 - 9iT - 83T^{2} \)
89 \( 1 - 15T + 89T^{2} \)
97 \( 1 + 10iT - 97T^{2} \)
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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

−11.92881269295989137345621351552, −10.47158512922332734982283242374, −9.568685522922711961902715967641, −8.585608861462356437225398783181, −7.64782974353942160188456327130, −6.81224159074683869205437025071, −5.92173721221202844095837243996, −4.75532718895897824441470639549, −3.38090030408863777519563102748, −1.39931523367926616697580766515, 1.38643737698553398363216844504, 3.22476268795815471130630124211, 4.22140969165292187868312288693, 5.12974229193952936245223353764, 6.67972868613410665024264603752, 7.66578944834140103048419730306, 9.137854964217652677403368906993, 9.579009428886797063318045648029, 10.47171892529940308758171697404, 11.43542080794687841361161320888

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