Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.877 + 0.480i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.91·2-s − 1.73i·3-s + 4.51·4-s − 2.23i·5-s − 5.05i·6-s + (6.13 + 3.36i)7-s + 1.49·8-s − 2.99·9-s − 6.52i·10-s − 2.58·11-s − 7.81i·12-s − 0.0498i·13-s + (17.9 + 9.80i)14-s − 3.87·15-s − 13.6·16-s + 14.2i·17-s + ⋯
L(s)  = 1  + 1.45·2-s − 0.577i·3-s + 1.12·4-s − 0.447i·5-s − 0.842i·6-s + (0.877 + 0.480i)7-s + 0.186·8-s − 0.333·9-s − 0.652i·10-s − 0.235·11-s − 0.651i·12-s − 0.00383i·13-s + (1.27 + 0.700i)14-s − 0.258·15-s − 0.855·16-s + 0.835i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 + 0.480i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.877 + 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.877 + 0.480i$
motivic weight  =  \(2\)
character  :  $\chi_{105} (76, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1),\ 0.877 + 0.480i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(2.61552 - 0.669209i\)
\(L(\frac12)\)  \(\approx\)  \(2.61552 - 0.669209i\)
\(L(2)\)   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 \( 1 + (-6.13 - 3.36i)T \)
good2 \( 1 - 2.91T + 4T^{2} \)
11 \( 1 + 2.58T + 121T^{2} \)
13 \( 1 + 0.0498iT - 169T^{2} \)
17 \( 1 - 14.2iT - 289T^{2} \)
19 \( 1 - 14.9iT - 361T^{2} \)
23 \( 1 + 22.2T + 529T^{2} \)
29 \( 1 - 17.4T + 841T^{2} \)
31 \( 1 + 6.36iT - 961T^{2} \)
37 \( 1 + 7.14T + 1.36e3T^{2} \)
41 \( 1 + 74.5iT - 1.68e3T^{2} \)
43 \( 1 - 79.2T + 1.84e3T^{2} \)
47 \( 1 + 81.3iT - 2.20e3T^{2} \)
53 \( 1 + 67.1T + 2.80e3T^{2} \)
59 \( 1 + 4.33iT - 3.48e3T^{2} \)
61 \( 1 - 109. iT - 3.72e3T^{2} \)
67 \( 1 + 49.1T + 4.48e3T^{2} \)
71 \( 1 - 97.3T + 5.04e3T^{2} \)
73 \( 1 + 116. iT - 5.32e3T^{2} \)
79 \( 1 - 98.8T + 6.24e3T^{2} \)
83 \( 1 + 112. iT - 6.88e3T^{2} \)
89 \( 1 - 77.7iT - 7.92e3T^{2} \)
97 \( 1 - 109. iT - 9.40e3T^{2} \)
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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

−13.49225629617485456329823113971, −12.33283705395250512168121518683, −12.03237573588615498671913089461, −10.68419251034872665528258923396, −8.843745581626272518182906376475, −7.75148960288030811568524955352, −6.14816823846782935165213987763, −5.28510065142993099830185198546, −3.97083947266056149043176909940, −2.11229438753656113950831550964, 2.79236252205494713049229162049, 4.24248050260734257194883325928, 5.10203877317179189996275216120, 6.45805860672701688227044896277, 7.85685078396426869534738678567, 9.478330699815161498734812695942, 10.88531378322602401728313321504, 11.52360772935150779691917575981, 12.71567616974641150364339479621, 14.04912674466329106294119606479

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