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Properties

Label 2-105-35.33-c1-0-1
Degree $2$
Conductor $105$
Sign $-0.381 - 0.924i$
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  + (0.602 + 2.24i)2-s + (0.965 + 0.258i)3-s + (−2.95 + 1.70i)4-s + (−1.28 − 1.82i)5-s + 2.32i·6-s + (0.519 + 2.59i)7-s + (−2.33 − 2.33i)8-s + (0.866 + 0.499i)9-s + (3.33 − 3.99i)10-s + (−1.76 − 3.05i)11-s + (−3.30 + 0.884i)12-s + (4.49 − 4.49i)13-s + (−5.51 + 2.73i)14-s + (−0.767 − 2.10i)15-s + (0.421 − 0.729i)16-s + (0.481 − 1.79i)17-s + ⋯
L(s)  = 1  + (0.425 + 1.58i)2-s + (0.557 + 0.149i)3-s + (−1.47 + 0.854i)4-s + (−0.574 − 0.818i)5-s + 0.950i·6-s + (0.196 + 0.980i)7-s + (−0.824 − 0.824i)8-s + (0.288 + 0.166i)9-s + (1.05 − 1.26i)10-s + (−0.531 − 0.921i)11-s + (−0.952 + 0.255i)12-s + (1.24 − 1.24i)13-s + (−1.47 + 0.730i)14-s + (−0.198 − 0.542i)15-s + (0.105 − 0.182i)16-s + (0.116 − 0.436i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.381 - 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.381 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.381 - 0.924i$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1/2),\ -0.381 - 0.924i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.720624 + 1.07725i\)
\(L(\frac12)\) \(\approx\) \(0.720624 + 1.07725i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.965 - 0.258i)T \)
5 \( 1 + (1.28 + 1.82i)T \)
7 \( 1 + (-0.519 - 2.59i)T \)
good2 \( 1 + (-0.602 - 2.24i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (1.76 + 3.05i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.49 + 4.49i)T - 13iT^{2} \)
17 \( 1 + (-0.481 + 1.79i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.0699 - 0.121i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.72 - 0.997i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 2.01iT - 29T^{2} \)
31 \( 1 + (4.56 - 2.63i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.50 + 5.61i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 0.903iT - 41T^{2} \)
43 \( 1 + (-2.38 - 2.38i)T + 43iT^{2} \)
47 \( 1 + (2.38 - 0.639i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (0.726 - 2.71i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-3.15 - 5.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (8.69 + 5.01i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-10.3 - 2.77i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 5.09T + 71T^{2} \)
73 \( 1 + (-9.04 - 2.42i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-7.30 - 4.21i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.37 + 7.37i)T - 83iT^{2} \)
89 \( 1 + (-1.75 + 3.03i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.70 + 8.70i)T + 97iT^{2} \)
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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

−14.28807180087816921163064585512, −13.32728094028858731493069652407, −12.53615902573870203565533846852, −11.00281081519332042459050834412, −9.050550606549139632699791770007, −8.363059925524310267182745194610, −7.71568422257950267388210129110, −5.91012909381126980110719178111, −5.16472330321037241105865725681, −3.56822427576357468106290820030, 1.94696912411311418490306036974, 3.59364036884654689035602462713, 4.33935087873526396450976443948, 6.78042405126188177513731624112, 8.036216842097885979000585195666, 9.610511905606179855263690831874, 10.55732956926830706104303113237, 11.27869196024513402580458414032, 12.29689051406933216743861816659, 13.44526656710771794827809642594

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