Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7^{2} $
Sign $0.326 + 0.945i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0611 − 0.228i)2-s + (0.965 + 0.258i)3-s + (1.68 − 0.972i)4-s + (1.04 − 1.97i)5-s − 0.236i·6-s + (−0.658 − 0.658i)8-s + (0.866 + 0.499i)9-s + (−0.515 − 0.117i)10-s + (−1.99 − 3.45i)11-s + (1.87 − 0.503i)12-s + (−0.500 + 0.500i)13-s + (1.52 − 1.63i)15-s + (1.83 − 3.17i)16-s + (0.614 − 2.29i)17-s + (0.0611 − 0.228i)18-s + (−3.60 + 6.25i)19-s + ⋯
L(s)  = 1  + (−0.0432 − 0.161i)2-s + (0.557 + 0.149i)3-s + (0.841 − 0.486i)4-s + (0.467 − 0.884i)5-s − 0.0964i·6-s + (−0.232 − 0.232i)8-s + (0.288 + 0.166i)9-s + (−0.162 − 0.0371i)10-s + (−0.600 − 1.04i)11-s + (0.542 − 0.145i)12-s + (−0.138 + 0.138i)13-s + (0.392 − 0.423i)15-s + (0.458 − 0.794i)16-s + (0.148 − 0.556i)17-s + (0.0144 − 0.0537i)18-s + (−0.828 + 1.43i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.326 + 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.326 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
\( \varepsilon \)  =  $0.326 + 0.945i$
motivic weight  =  \(1\)
character  :  $\chi_{735} (313, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 735,\ (\ :1/2),\ 0.326 + 0.945i)\)
\(L(1)\)  \(\approx\)  \(1.82242 - 1.29857i\)
\(L(\frac12)\)  \(\approx\)  \(1.82242 - 1.29857i\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + (-0.965 - 0.258i)T \)
5 \( 1 + (-1.04 + 1.97i)T \)
7 \( 1 \)
good2 \( 1 + (0.0611 + 0.228i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (1.99 + 3.45i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.500 - 0.500i)T - 13iT^{2} \)
17 \( 1 + (-0.614 + 2.29i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (3.60 - 6.25i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-7.04 + 1.88i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 3.65iT - 29T^{2} \)
31 \( 1 + (4.27 - 2.46i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.106 - 0.399i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 7.63iT - 41T^{2} \)
43 \( 1 + (-3.65 - 3.65i)T + 43iT^{2} \)
47 \( 1 + (0.417 - 0.111i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.97 + 7.37i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-3.05 - 5.29i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.15 - 3.55i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.28 + 0.345i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 1.19T + 71T^{2} \)
73 \( 1 + (-1.88 - 0.506i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-7.48 - 4.32i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-11.9 + 11.9i)T - 83iT^{2} \)
89 \( 1 + (-3.91 + 6.77i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.43 + 7.43i)T + 97iT^{2} \)
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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

−10.25896351686179184333657631658, −9.357992437234640225684677865934, −8.600281719964671384147819463498, −7.76382818890873047510902439903, −6.62759745063098631412269264528, −5.69165056350705128171257067209, −4.90767581925933277345207610642, −3.43015872580151572693701726448, −2.36759355192629916168651605412, −1.14791690118551016948437351597, 2.09990195003607746699638570706, 2.66633520587242778898720570119, 3.82574731302468326337874862082, 5.28016210804028446197931458846, 6.47931558962248240559015871127, 7.15161782317007432970677508548, 7.67430107228630524052951876501, 8.805968763168811241602026085063, 9.683146745001442630717460867800, 10.71527061739897197904060703637

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