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Properties

Label 2-105-105.59-c1-0-5
Degree $2$
Conductor $105$
Sign $0.922 + 0.386i$
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  + (−0.322 − 0.558i)2-s + (1.69 + 0.358i)3-s + (0.792 − 1.37i)4-s + (0.550 + 2.16i)5-s + (−0.346 − 1.06i)6-s + (−2.64 − 0.105i)7-s − 2.31·8-s + (2.74 + 1.21i)9-s + (1.03 − 1.00i)10-s + (−3.51 − 2.02i)11-s + (1.83 − 2.04i)12-s + 4.21·13-s + (0.793 + 1.51i)14-s + (0.155 + 3.86i)15-s + (−0.839 − 1.45i)16-s + (−1.88 − 1.08i)17-s + ⋯
L(s)  = 1  + (−0.227 − 0.394i)2-s + (0.978 + 0.206i)3-s + (0.396 − 0.685i)4-s + (0.246 + 0.969i)5-s + (−0.141 − 0.433i)6-s + (−0.999 − 0.0397i)7-s − 0.817·8-s + (0.914 + 0.404i)9-s + (0.326 − 0.318i)10-s + (−1.05 − 0.611i)11-s + (0.529 − 0.589i)12-s + 1.16·13-s + (0.212 + 0.403i)14-s + (0.0401 + 0.999i)15-s + (−0.209 − 0.363i)16-s + (−0.457 − 0.263i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 + 0.386i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.922 + 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.922 + 0.386i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1/2),\ 0.922 + 0.386i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18081 - 0.237161i\)
\(L(\frac12)\) \(\approx\) \(1.18081 - 0.237161i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.69 - 0.358i)T \)
5 \( 1 + (-0.550 - 2.16i)T \)
7 \( 1 + (2.64 + 0.105i)T \)
good2 \( 1 + (0.322 + 0.558i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (3.51 + 2.02i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 4.21T + 13T^{2} \)
17 \( 1 + (1.88 + 1.08i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.87 - 2.23i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.322 - 0.558i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.16iT - 29T^{2} \)
31 \( 1 + (0.339 + 0.195i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.69 + 2.13i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.27T + 41T^{2} \)
43 \( 1 - 6.54iT - 43T^{2} \)
47 \( 1 + (-6.75 + 3.90i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.60 + 6.23i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.66 + 9.80i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.05 + 3.49i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.56 + 4.36i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.13iT - 71T^{2} \)
73 \( 1 + (2.61 - 4.53i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.87 + 3.24i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 5.27iT - 83T^{2} \)
89 \( 1 + (-0.447 - 0.774i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 3.89T + 97T^{2} \)
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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

−13.72366821589172743915997749166, −12.95069541313786212337099060608, −11.13923559732609138146047293218, −10.44595114158278908816783919758, −9.652144774608468766433858135616, −8.457562759169854253451041540192, −6.91840129132740711807127738652, −5.88821283440211278284221118398, −3.49070506636859010470744956243, −2.40575844550791626752710636669, 2.51562417070333791760735785849, 4.07380185465940992718650235018, 6.14683189651190451471463461317, 7.34273207579114944930531081076, 8.499247797569248002964022813388, 9.062909525196084954954542146958, 10.39382933246475342275046722995, 12.17714277382256251227036995970, 13.09735329503993149924878561216, 13.35990057158874055017491481298

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