Properties

Degree 2
Conductor $ 2 \cdot 5^{2} \cdot 7 $
Sign $0.447 + 0.894i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s − 2i·3-s − 4-s + 2·6-s i·7-s i·8-s − 9-s + 2i·12-s − 4i·13-s + 14-s + 16-s − 6i·17-s i·18-s − 2·19-s − 2·21-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.15i·3-s − 0.5·4-s + 0.816·6-s − 0.377i·7-s − 0.353i·8-s − 0.333·9-s + 0.577i·12-s − 1.10i·13-s + 0.267·14-s + 0.250·16-s − 1.45i·17-s − 0.235i·18-s − 0.458·19-s − 0.436·21-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.447 + 0.894i$
motivic weight  =  \(1\)
character  :  $\chi_{350} (99, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 350,\ (\ :1/2),\ 0.447 + 0.894i)\)
\(L(1)\)  \(\approx\)  \(1.00849 - 0.623283i\)
\(L(\frac12)\)  \(\approx\)  \(1.00849 - 0.623283i\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 \)
7 \( 1 + iT \)
good3 \( 1 + 2iT - 3T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 8iT - 43T^{2} \)
47 \( 1 - 12iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 - 6T + 59T^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 - 10iT - 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−11.46975453226636471827058596761, −10.33953066651889968860924777574, −9.296527828953851678438374022882, −8.104838269498584631069763936761, −7.48736582630619125405222246396, −6.70228708980095922945847101253, −5.70856463881718166521226143434, −4.47992619641238253053421396260, −2.78433191336991662037945800384, −0.875332837139987971235460555126, 2.01314549923338325550520851977, 3.63535422454790820219193927071, 4.35574163478167456399141262308, 5.45836858458498125749419368293, 6.75392696237221147300912939967, 8.393707496562347182363489127918, 9.049577387658908266269627866472, 10.00163421698977404390334482327, 10.60105785382614769177464695896, 11.48583316339616217251470155317

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