Properties

Degree 2
Conductor $ 3 \cdot 5 \cdot 7 $
Sign $0.639 - 0.769i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.298 + 0.0799i)2-s + (1.15 + 1.29i)3-s + (−1.64 + 0.952i)4-s + (1.56 − 1.59i)5-s + (−0.447 − 0.292i)6-s + (0.951 + 2.46i)7-s + (0.852 − 0.852i)8-s + (−0.333 + 2.98i)9-s + (−0.340 + 0.600i)10-s + (−0.660 + 0.381i)11-s + (−3.13 − 1.02i)12-s + (−2.27 − 2.27i)13-s + (−0.481 − 0.660i)14-s + (3.86 + 0.184i)15-s + (1.71 − 2.97i)16-s + (1.25 − 4.69i)17-s + ⋯
L(s)  = 1  + (−0.210 + 0.0565i)2-s + (0.666 + 0.745i)3-s + (−0.824 + 0.476i)4-s + (0.701 − 0.712i)5-s + (−0.182 − 0.119i)6-s + (0.359 + 0.933i)7-s + (0.301 − 0.301i)8-s + (−0.111 + 0.993i)9-s + (−0.107 + 0.189i)10-s + (−0.199 + 0.114i)11-s + (−0.904 − 0.297i)12-s + (−0.629 − 0.629i)13-s + (−0.128 − 0.176i)14-s + (0.998 + 0.0476i)15-s + (0.429 − 0.744i)16-s + (0.305 − 1.13i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.639 - 0.769i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.639 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(105\)    =    \(3 \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.639 - 0.769i$
motivic weight  =  \(1\)
character  :  $\chi_{105} (23, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 105,\ (\ :1/2),\ 0.639 - 0.769i)$
$L(1)$  $\approx$  $0.951699 + 0.446543i$
$L(\frac12)$  $\approx$  $0.951699 + 0.446543i$
$L(\frac{3}{2})$   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 \( 1 + (-1.15 - 1.29i)T \)
5 \( 1 + (-1.56 + 1.59i)T \)
7 \( 1 + (-0.951 - 2.46i)T \)
good2 \( 1 + (0.298 - 0.0799i)T + (1.73 - i)T^{2} \)
11 \( 1 + (0.660 - 0.381i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.27 + 2.27i)T + 13iT^{2} \)
17 \( 1 + (-1.25 + 4.69i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.41 - 0.818i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.98 + 7.39i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 4.94T + 29T^{2} \)
31 \( 1 + (-2.96 - 5.13i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.915 + 3.41i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 4.35iT - 41T^{2} \)
43 \( 1 + (-2.69 - 2.69i)T + 43iT^{2} \)
47 \( 1 + (4.14 - 1.10i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-6.71 - 1.79i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-3.84 - 6.65i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.19 - 3.80i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.0471 + 0.0126i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 12.4iT - 71T^{2} \)
73 \( 1 + (-0.359 + 1.34i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (3.66 + 2.11i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.05 - 5.05i)T - 83iT^{2} \)
89 \( 1 + (0.453 - 0.785i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.73 + 3.73i)T - 97iT^{2} \)
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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

−13.99920429596656099167880159566, −12.96059429743635006630608208119, −12.07605849668141608362615803380, −10.27256839377607404031674377481, −9.422852579145509056349884036700, −8.697365105452968453904277484431, −7.78631863216822720128372315683, −5.40212568551258648384813353834, −4.61848967732270377794003124876, −2.72083510051971430192810921346, 1.74454804715826243937081266219, 3.81663674748601830592254861361, 5.67740741984274556196796251508, 7.07133504332943469769667803083, 8.087691135656367954286586490834, 9.447252873630185922125345976477, 10.14863833897985606208511081011, 11.42197141530771500397324141210, 13.05699419930619686874248991217, 13.71666647681976269727169916555

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