Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7^{2} $
Sign $-0.511 - 0.859i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.681 + 2.54i)2-s + (0.965 − 0.258i)3-s + (−4.26 − 2.46i)4-s + (2.18 + 0.478i)5-s + 2.63i·6-s + (5.45 − 5.45i)8-s + (0.866 − 0.499i)9-s + (−2.70 + 5.22i)10-s + (0.731 − 1.26i)11-s + (−4.76 − 1.27i)12-s + (0.887 + 0.887i)13-s + (2.23 − 0.103i)15-s + (5.22 + 9.04i)16-s + (−0.770 − 2.87i)17-s + (0.681 + 2.54i)18-s + (1.97 + 3.42i)19-s + ⋯
L(s)  = 1  + (−0.481 + 1.79i)2-s + (0.557 − 0.149i)3-s + (−2.13 − 1.23i)4-s + (0.976 + 0.213i)5-s + 1.07i·6-s + (1.92 − 1.92i)8-s + (0.288 − 0.166i)9-s + (−0.855 + 1.65i)10-s + (0.220 − 0.381i)11-s + (−1.37 − 0.368i)12-s + (0.246 + 0.246i)13-s + (0.576 − 0.0267i)15-s + (1.30 + 2.26i)16-s + (−0.186 − 0.697i)17-s + (0.160 + 0.599i)18-s + (0.454 + 0.786i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.511 - 0.859i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.511 - 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(735\)    =    \(3 \cdot 5 \cdot 7^{2}\)
\( \varepsilon \)  =  $-0.511 - 0.859i$
motivic weight  =  \(1\)
character  :  $\chi_{735} (472, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 735,\ (\ :1/2),\ -0.511 - 0.859i)\)
\(L(1)\)  \(\approx\)  \(0.772147 + 1.35761i\)
\(L(\frac12)\)  \(\approx\)  \(0.772147 + 1.35761i\)
\(L(\frac{3}{2})\)   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 \( 1 + (-2.18 - 0.478i)T \)
7 \( 1 \)
good2 \( 1 + (0.681 - 2.54i)T + (-1.73 - i)T^{2} \)
11 \( 1 + (-0.731 + 1.26i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.887 - 0.887i)T + 13iT^{2} \)
17 \( 1 + (0.770 + 2.87i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.97 - 3.42i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.64 - 1.51i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 5.18iT - 29T^{2} \)
31 \( 1 + (-5.28 - 3.05i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.825 + 3.08i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 0.769iT - 41T^{2} \)
43 \( 1 + (5.18 - 5.18i)T - 43iT^{2} \)
47 \( 1 + (11.7 + 3.13i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.199 + 0.743i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-1.59 + 2.76i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.23 + 0.710i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.10 + 2.17i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 7.62T + 71T^{2} \)
73 \( 1 + (9.30 - 2.49i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-3.91 + 2.26i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.75 + 6.75i)T + 83iT^{2} \)
89 \( 1 + (0.599 + 1.03i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.68 - 8.68i)T - 97iT^{2} \)
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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

−10.06810504445726272590810077295, −9.500630877995183153958965748467, −8.788761990864513770245233199010, −8.094658126496667643268774657297, −7.00511738147410948526752633364, −6.57964296606393523405866275655, −5.56526949445614087064634109942, −4.79600158320141568453804509069, −3.22429315976426993191090711586, −1.33883398695939114863828061841, 1.13598076038366903758862658819, 2.24598451968914789074927891398, 3.06587048717464498499223157110, 4.24545078177122742784998718265, 5.13200101662924027985810304061, 6.64489074968844526921349087770, 8.100099282444350747154729321419, 8.753899375728644503727603422305, 9.554754631685216221994783140934, 9.984666998740503159232287873498

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