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Properties

Label 2-350-5.4-c3-0-15
Degree $2$
Conductor $350$
Sign $0.894 - 0.447i$
Analytic cond. $20.6506$
Root an. cond. $4.54430$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2i·2-s + 4i·3-s − 4·4-s − 8·6-s − 7i·7-s − 8i·8-s + 11·9-s + 5·11-s − 16i·12-s − 82i·13-s + 14·14-s + 16·16-s − 12i·17-s + 22i·18-s + 42·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.769i·3-s − 0.5·4-s − 0.544·6-s − 0.377i·7-s − 0.353i·8-s + 0.407·9-s + 0.137·11-s − 0.384i·12-s − 1.74i·13-s + 0.267·14-s + 0.250·16-s − 0.171i·17-s + 0.288i·18-s + 0.507·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(20.6506\)
Root analytic conductor: \(4.54430\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :3/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.754314812\)
\(L(\frac12)\) \(\approx\) \(1.754314812\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2iT \)
5 \( 1 \)
7 \( 1 + 7iT \)
good3 \( 1 - 4iT - 27T^{2} \)
11 \( 1 - 5T + 1.33e3T^{2} \)
13 \( 1 + 82iT - 2.19e3T^{2} \)
17 \( 1 + 12iT - 4.91e3T^{2} \)
19 \( 1 - 42T + 6.85e3T^{2} \)
23 \( 1 + 175iT - 1.21e4T^{2} \)
29 \( 1 + T + 2.43e4T^{2} \)
31 \( 1 - 226T + 2.97e4T^{2} \)
37 \( 1 + 19iT - 5.06e4T^{2} \)
41 \( 1 - 16T + 6.89e4T^{2} \)
43 \( 1 + 281iT - 7.95e4T^{2} \)
47 \( 1 - 334iT - 1.03e5T^{2} \)
53 \( 1 - 398iT - 1.48e5T^{2} \)
59 \( 1 + 106T + 2.05e5T^{2} \)
61 \( 1 - 48T + 2.26e5T^{2} \)
67 \( 1 - 483iT - 3.00e5T^{2} \)
71 \( 1 + 15T + 3.57e5T^{2} \)
73 \( 1 + 1.04e3iT - 3.89e5T^{2} \)
79 \( 1 - 1.25e3T + 4.93e5T^{2} \)
83 \( 1 + 758iT - 5.71e5T^{2} \)
89 \( 1 + 86T + 7.04e5T^{2} \)
97 \( 1 - 710iT - 9.12e5T^{2} \)
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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

−10.66560709921042268840084654607, −10.26925474773874803237329613024, −9.290509715300972407834825053010, −8.228567947035707006389265355201, −7.39117894098126002599413150119, −6.25705035058007297310911286034, −5.13235829345022881097627823780, −4.28160841356787353520963141142, −3.05792702224212663854573183874, −0.72845727608546573999796201431, 1.24784989889817536008594891379, 2.18692095830498861440865461940, 3.71589610073648110915943104635, 4.86745854549275649088650486958, 6.26373415691182703731166237286, 7.16949694773755817967264224995, 8.232379886272797760183249487025, 9.330204703659582403276758708952, 9.948296661772883470218523104855, 11.38989181369026411067706938860

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