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Properties

Label 2-105-7.6-c2-0-4
Degree $2$
Conductor $105$
Sign $0.349 + 0.936i$
Analytic cond. $2.86104$
Root an. cond. $1.69146$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.80·2-s − 1.73i·3-s + 10.4·4-s − 2.23i·5-s + 6.58i·6-s + (2.44 + 6.55i)7-s − 24.6·8-s − 2.99·9-s + 8.50i·10-s + 14.4·11-s − 18.1i·12-s − 16.9i·13-s + (−9.30 − 24.9i)14-s − 3.87·15-s + 51.8·16-s − 13.0i·17-s + ⋯
L(s)  = 1  − 1.90·2-s − 0.577i·3-s + 2.61·4-s − 0.447i·5-s + 1.09i·6-s + (0.349 + 0.936i)7-s − 3.07·8-s − 0.333·9-s + 0.850i·10-s + 1.31·11-s − 1.51i·12-s − 1.30i·13-s + (−0.664 − 1.78i)14-s − 0.258·15-s + 3.23·16-s − 0.765i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.349 + 0.936i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.349 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.349 + 0.936i$
Analytic conductor: \(2.86104\)
Root analytic conductor: \(1.69146\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1),\ 0.349 + 0.936i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.497459 - 0.345394i\)
\(L(\frac12)\) \(\approx\) \(0.497459 - 0.345394i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.73iT \)
5 \( 1 + 2.23iT \)
7 \( 1 + (-2.44 - 6.55i)T \)
good2 \( 1 + 3.80T + 4T^{2} \)
11 \( 1 - 14.4T + 121T^{2} \)
13 \( 1 + 16.9iT - 169T^{2} \)
17 \( 1 + 13.0iT - 289T^{2} \)
19 \( 1 + 18.6iT - 361T^{2} \)
23 \( 1 - 10.3T + 529T^{2} \)
29 \( 1 + 13.7T + 841T^{2} \)
31 \( 1 + 42.4iT - 961T^{2} \)
37 \( 1 - 28.7T + 1.36e3T^{2} \)
41 \( 1 - 28.8iT - 1.68e3T^{2} \)
43 \( 1 + 5.84T + 1.84e3T^{2} \)
47 \( 1 + 10.5iT - 2.20e3T^{2} \)
53 \( 1 - 81.9T + 2.80e3T^{2} \)
59 \( 1 + 35.1iT - 3.48e3T^{2} \)
61 \( 1 - 68.4iT - 3.72e3T^{2} \)
67 \( 1 - 47.4T + 4.48e3T^{2} \)
71 \( 1 - 47.6T + 5.04e3T^{2} \)
73 \( 1 - 125. iT - 5.32e3T^{2} \)
79 \( 1 + 129.T + 6.24e3T^{2} \)
83 \( 1 - 42.6iT - 6.88e3T^{2} \)
89 \( 1 + 25.5iT - 7.92e3T^{2} \)
97 \( 1 + 28.7iT - 9.40e3T^{2} \)
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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

−12.96058860160822868982856382237, −11.77253996352174374965358276139, −11.26885783032762230069715831532, −9.694682544418985415887097712865, −8.932519613579912241784705849607, −8.088740004223996361113819315546, −6.99154190300478446127319596887, −5.73245182492781617760377294435, −2.54524952247243406350548104096, −0.914502784216565748829261743206, 1.57722150818967912378539568425, 3.81449134162338526634134408736, 6.38419932816666072798134056969, 7.25372738872736290176413272560, 8.532083305605571199749227397265, 9.435761381238995222141767444526, 10.36876249508753657705046810584, 11.13832836727804306069342850702, 11.99410170717044151615831830584, 14.19827333710249341962283580367

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