Properties

Label 2-105-21.20-c1-0-8
Degree $2$
Conductor $105$
Sign $0.0406 + 0.999i$
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  − 0.792i·2-s + (−1.18 − 1.26i)3-s + 1.37·4-s + 5-s + (−1 + 0.939i)6-s + (−2 − 1.73i)7-s − 2.67i·8-s + (−0.186 + 2.99i)9-s − 0.792i·10-s − 2.52i·11-s + (−1.62 − 1.73i)12-s + 4.10i·13-s + (−1.37 + 1.58i)14-s + (−1.18 − 1.26i)15-s + 0.627·16-s + 4.37·17-s + ⋯
L(s)  = 1  − 0.560i·2-s + (−0.684 − 0.728i)3-s + 0.686·4-s + 0.447·5-s + (−0.408 + 0.383i)6-s + (−0.755 − 0.654i)7-s − 0.944i·8-s + (−0.0620 + 0.998i)9-s − 0.250i·10-s − 0.761i·11-s + (−0.469 − 0.499i)12-s + 1.13i·13-s + (−0.366 + 0.423i)14-s + (−0.306 − 0.325i)15-s + 0.156·16-s + 1.06·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0406 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0406 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.0406 + 0.999i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1/2),\ 0.0406 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.713932 - 0.685498i\)
\(L(\frac12)\) \(\approx\) \(0.713932 - 0.685498i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.18 + 1.26i)T \)
5 \( 1 - T \)
7 \( 1 + (2 + 1.73i)T \)
good2 \( 1 + 0.792iT - 2T^{2} \)
11 \( 1 + 2.52iT - 11T^{2} \)
13 \( 1 - 4.10iT - 13T^{2} \)
17 \( 1 - 4.37T + 17T^{2} \)
19 \( 1 - 3.46iT - 19T^{2} \)
23 \( 1 - 8.51iT - 23T^{2} \)
29 \( 1 - 0.939iT - 29T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 + 6.74T + 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 4.74T + 43T^{2} \)
47 \( 1 + 1.62T + 47T^{2} \)
53 \( 1 + 1.87iT - 53T^{2} \)
59 \( 1 + 8.74T + 59T^{2} \)
61 \( 1 + 6.92iT - 61T^{2} \)
67 \( 1 - 4.74T + 67T^{2} \)
71 \( 1 - 0.294iT - 71T^{2} \)
73 \( 1 + 6.92iT - 73T^{2} \)
79 \( 1 + 2.37T + 79T^{2} \)
83 \( 1 + 17.4T + 83T^{2} \)
89 \( 1 - 14.7T + 89T^{2} \)
97 \( 1 - 11.0iT - 97T^{2} \)
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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

−13.35583342341271489832251525232, −12.29653979788912844549062169392, −11.51882845345461886279610256714, −10.54252068835282817091018180877, −9.592083117807769712215864373864, −7.65702002748354059575768075344, −6.66631235715714539946724161788, −5.72745510780103229530670577421, −3.46983025312184155536101997926, −1.55410476365347686804164752499, 2.91819286725912666469207734611, 5.09420919108976956957106794419, 6.01296208264307035323013516221, 6.99914200406531654672298149068, 8.635526363109768257033067448781, 9.970339824414030403122522356040, 10.64762928898812412792599821727, 12.04498539489704649177272081399, 12.68333510364610546997142348472, 14.45762990103823113045855005150

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