Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.349 - 0.936i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.80·2-s + 1.73i·3-s + 10.4·4-s + 2.23i·5-s − 6.58i·6-s + (2.44 − 6.55i)7-s − 24.6·8-s − 2.99·9-s − 8.50i·10-s + 14.4·11-s + 18.1i·12-s + 16.9i·13-s + (−9.30 + 24.9i)14-s − 3.87·15-s + 51.8·16-s + 13.0i·17-s + ⋯
L(s)  = 1  − 1.90·2-s + 0.577i·3-s + 2.61·4-s + 0.447i·5-s − 1.09i·6-s + (0.349 − 0.936i)7-s − 3.07·8-s − 0.333·9-s − 0.850i·10-s + 1.31·11-s + 1.51i·12-s + 1.30i·13-s + (−0.664 + 1.78i)14-s − 0.258·15-s + 3.23·16-s + 0.765i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.349 - 0.936i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.349 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.349 - 0.936i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (76, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ 0.349 - 0.936i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.497459 + 0.345394i\)
\(L(\frac12)\)  \(\approx\)  \(0.497459 + 0.345394i\)
\(L(2)\)   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 \( 1 - 1.73iT \)
5 \( 1 - 2.23iT \)
7 \( 1 + (-2.44 + 6.55i)T \)
good2 \( 1 + 3.80T + 4T^{2} \)
11 \( 1 - 14.4T + 121T^{2} \)
13 \( 1 - 16.9iT - 169T^{2} \)
17 \( 1 - 13.0iT - 289T^{2} \)
19 \( 1 - 18.6iT - 361T^{2} \)
23 \( 1 - 10.3T + 529T^{2} \)
29 \( 1 + 13.7T + 841T^{2} \)
31 \( 1 - 42.4iT - 961T^{2} \)
37 \( 1 - 28.7T + 1.36e3T^{2} \)
41 \( 1 + 28.8iT - 1.68e3T^{2} \)
43 \( 1 + 5.84T + 1.84e3T^{2} \)
47 \( 1 - 10.5iT - 2.20e3T^{2} \)
53 \( 1 - 81.9T + 2.80e3T^{2} \)
59 \( 1 - 35.1iT - 3.48e3T^{2} \)
61 \( 1 + 68.4iT - 3.72e3T^{2} \)
67 \( 1 - 47.4T + 4.48e3T^{2} \)
71 \( 1 - 47.6T + 5.04e3T^{2} \)
73 \( 1 + 125. iT - 5.32e3T^{2} \)
79 \( 1 + 129.T + 6.24e3T^{2} \)
83 \( 1 + 42.6iT - 6.88e3T^{2} \)
89 \( 1 - 25.5iT - 7.92e3T^{2} \)
97 \( 1 - 28.7iT - 9.40e3T^{2} \)
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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

−14.19827333710249341962283580367, −11.99410170717044151615831830584, −11.13832836727804306069342850702, −10.36876249508753657705046810584, −9.435761381238995222141767444526, −8.532083305605571199749227397265, −7.25372738872736290176413272560, −6.38419932816666072798134056969, −3.81449134162338526634134408736, −1.57722150818967912378539568425, 0.914502784216565748829261743206, 2.54524952247243406350548104096, 5.73245182492781617760377294435, 6.99154190300478446127319596887, 8.088740004223996361113819315546, 8.932519613579912241784705849607, 9.694682544418985415887097712865, 11.26885783032762230069715831532, 11.77253996352174374965358276139, 12.96058860160822868982856382237

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