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Properties

Label 2-105-105.32-c1-0-7
Degree $2$
Conductor $105$
Sign $0.639 + 0.769i$
Analytic cond. $0.838429$
Root an. cond. $0.915657$
Motivic weight $1$
Arithmetic yes
Rational no
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

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.639 + 0.769i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1/2),\ 0.639 + 0.769i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.951699 - 0.446543i\)
\(L(\frac12)\) \(\approx\) \(0.951699 - 0.446543i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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