Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.998 + 0.0607i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.340 − 1.26i)2-s + (0.224 + 1.71i)3-s + (0.236 + 0.136i)4-s + (−1.25 + 1.85i)5-s + (2.25 + 0.299i)6-s + (2.32 − 1.25i)7-s + (2.11 − 2.11i)8-s + (−2.89 + 0.769i)9-s + (1.92 + 2.21i)10-s + (−3.38 − 1.95i)11-s + (−0.181 + 0.436i)12-s + (−1.56 − 1.56i)13-s + (−0.807 − 3.38i)14-s + (−3.46 − 1.73i)15-s + (−1.69 − 2.92i)16-s + (2.58 − 0.693i)17-s + ⋯
L(s)  = 1  + (0.240 − 0.897i)2-s + (0.129 + 0.991i)3-s + (0.118 + 0.0681i)4-s + (−0.559 + 0.829i)5-s + (0.921 + 0.122i)6-s + (0.879 − 0.476i)7-s + (0.746 − 0.746i)8-s + (−0.966 + 0.256i)9-s + (0.609 + 0.701i)10-s + (−1.01 − 0.588i)11-s + (−0.0523 + 0.125i)12-s + (−0.434 − 0.434i)13-s + (−0.215 − 0.903i)14-s + (−0.894 − 0.447i)15-s + (−0.422 − 0.731i)16-s + (0.627 − 0.168i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0607i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0607i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.998 + 0.0607i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (53, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 105,\ (\ :1/2),\ 0.998 + 0.0607i)\)
\(L(1)\)  \(\approx\)  \(1.22657 - 0.0373146i\)
\(L(\frac12)\)  \(\approx\)  \(1.22657 - 0.0373146i\)
\(L(\frac{3}{2})\)   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 \( 1 + (-0.224 - 1.71i)T \)
5 \( 1 + (1.25 - 1.85i)T \)
7 \( 1 + (-2.32 + 1.25i)T \)
good2 \( 1 + (-0.340 + 1.26i)T + (-1.73 - i)T^{2} \)
11 \( 1 + (3.38 + 1.95i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.56 + 1.56i)T + 13iT^{2} \)
17 \( 1 + (-2.58 + 0.693i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.61 - 0.930i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.38 - 0.638i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 0.513T + 29T^{2} \)
31 \( 1 + (4.29 - 7.43i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (6.60 + 1.77i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 0.308iT - 41T^{2} \)
43 \( 1 + (-7.60 - 7.60i)T + 43iT^{2} \)
47 \( 1 + (-1.36 + 5.10i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (0.498 + 1.85i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.259 - 0.448i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.55 + 4.42i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.34 - 8.74i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 15.3iT - 71T^{2} \)
73 \( 1 + (-2.79 + 0.749i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-4.37 + 2.52i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (9.16 - 9.16i)T - 83iT^{2} \)
89 \( 1 + (-5.67 - 9.82i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.81 - 6.81i)T - 97iT^{2} \)
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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

−13.86960468559094906686018170345, −12.46099271854131731270015475483, −11.27052704194844822334282972088, −10.77880840524231199925538961088, −10.09072057733748490440179601708, −8.267064887130154162109985129140, −7.29148039382786147226254168483, −5.20345071290482051891539274991, −3.81941783381491378346168396626, −2.77732612026732877221391517755, 2.03146238300716129021761020825, 4.81715121360924810281503101819, 5.75470820309427466431828730814, 7.35151424084147504762270334992, 7.85328612111026418827327509830, 8.926251483029510006205350952309, 10.90583600566160765459394741909, 11.95525509040646853269615438380, 12.77813787151423467880683774779, 13.91467102477436123553490575835

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