Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.957 - 0.288i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.167 + 0.167i)2-s + (0.707 − 0.707i)3-s + 1.94i·4-s + (2.23 − 0.0836i)5-s + 0.236i·6-s + (−2.64 + 0.0627i)7-s + (−0.658 − 0.658i)8-s − 1.00i·9-s + (−0.359 + 0.387i)10-s + 3.98·11-s + (1.37 + 1.37i)12-s + (0.500 − 0.500i)13-s + (0.431 − 0.452i)14-s + (1.52 − 1.63i)15-s − 3.66·16-s + (−1.67 − 1.67i)17-s + ⋯
L(s)  = 1  + (−0.118 + 0.118i)2-s + (0.408 − 0.408i)3-s + 0.972i·4-s + (0.999 − 0.0373i)5-s + 0.0964i·6-s + (−0.999 + 0.0237i)7-s + (−0.232 − 0.232i)8-s − 0.333i·9-s + (−0.113 + 0.122i)10-s + 1.20·11-s + (0.396 + 0.396i)12-s + (0.138 − 0.138i)13-s + (0.115 − 0.120i)14-s + (0.392 − 0.423i)15-s − 0.917·16-s + (−0.407 − 0.407i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.957 - 0.288i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (13, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1/2),\ 0.957 - 0.288i)\)
\(L(1)\)  \(\approx\)  \(1.12400 + 0.165813i\)
\(L(\frac12)\)  \(\approx\)  \(1.12400 + 0.165813i\)
\(L(\frac{3}{2})\)   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 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (-2.23 + 0.0836i)T \)
7 \( 1 + (2.64 - 0.0627i)T \)
good2 \( 1 + (0.167 - 0.167i)T - 2iT^{2} \)
11 \( 1 - 3.98T + 11T^{2} \)
13 \( 1 + (-0.500 + 0.500i)T - 13iT^{2} \)
17 \( 1 + (1.67 + 1.67i)T + 17iT^{2} \)
19 \( 1 + 7.21T + 19T^{2} \)
23 \( 1 + (5.16 + 5.16i)T + 23iT^{2} \)
29 \( 1 - 3.65iT - 29T^{2} \)
31 \( 1 - 4.93iT - 31T^{2} \)
37 \( 1 + (-0.292 + 0.292i)T - 37iT^{2} \)
41 \( 1 + 7.63iT - 41T^{2} \)
43 \( 1 + (-3.65 - 3.65i)T + 43iT^{2} \)
47 \( 1 + (0.305 + 0.305i)T + 47iT^{2} \)
53 \( 1 + (-5.39 - 5.39i)T + 53iT^{2} \)
59 \( 1 - 6.10T + 59T^{2} \)
61 \( 1 - 7.11iT - 61T^{2} \)
67 \( 1 + (-0.944 + 0.944i)T - 67iT^{2} \)
71 \( 1 - 1.19T + 71T^{2} \)
73 \( 1 + (-1.38 + 1.38i)T - 73iT^{2} \)
79 \( 1 + 8.64iT - 79T^{2} \)
83 \( 1 + (11.9 - 11.9i)T - 83iT^{2} \)
89 \( 1 - 7.82T + 89T^{2} \)
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

−13.67181133343416460927075777799, −12.78444056436345442803068951196, −12.16485668597509931507013604426, −10.49088985786540512219479063112, −9.175543851258481321410862755918, −8.613644193172456770848011834828, −6.91764756835016110771546797330, −6.28514096937558453864195523927, −4.00126213727776887712711958504, −2.47473631653914824478364614979, 2.07813200094351623362048369923, 4.10105825201638125448174337138, 5.90377330477418653159297661449, 6.56547906066419388604444332290, 8.751251562101132845497298782935, 9.612024742923807442482917825286, 10.17380379886242029895542737338, 11.38012694445929410083328778366, 12.96934189602522066448103425824, 13.82359697871449011233308277260

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