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Properties

Label 2-105-21.17-c3-0-22
Degree $2$
Conductor $105$
Sign $0.324 + 0.945i$
Analytic cond. $6.19520$
Root an. cond. $2.48901$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.69 + 2.70i)2-s + (2.33 − 4.64i)3-s + (10.6 − 18.4i)4-s + (2.5 + 4.33i)5-s + (1.61 + 28.0i)6-s + (14.5 − 11.4i)7-s + 72.2i·8-s + (−16.0 − 21.6i)9-s + (−23.4 − 13.5i)10-s + (−38.4 − 22.2i)11-s + (−60.8 − 92.6i)12-s + 22.9i·13-s + (−36.9 + 93.2i)14-s + (25.9 − 1.49i)15-s + (−110. − 191. i)16-s + (32.3 − 55.9i)17-s + ⋯
L(s)  = 1  + (−1.65 + 0.957i)2-s + (0.449 − 0.893i)3-s + (1.33 − 2.30i)4-s + (0.223 + 0.387i)5-s + (0.110 + 1.91i)6-s + (0.784 − 0.620i)7-s + 3.19i·8-s + (−0.596 − 0.802i)9-s + (−0.741 − 0.428i)10-s + (−1.05 − 0.608i)11-s + (−1.46 − 2.22i)12-s + 0.489i·13-s + (−0.706 + 1.78i)14-s + (0.446 − 0.0256i)15-s + (−1.72 − 2.98i)16-s + (0.461 − 0.798i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.324 + 0.945i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.324 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $0.324 + 0.945i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ 0.324 + 0.945i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.591966 - 0.422902i\)
\(L(\frac12)\) \(\approx\) \(0.591966 - 0.422902i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.33 + 4.64i)T \)
5 \( 1 + (-2.5 - 4.33i)T \)
7 \( 1 + (-14.5 + 11.4i)T \)
good2 \( 1 + (4.69 - 2.70i)T + (4 - 6.92i)T^{2} \)
11 \( 1 + (38.4 + 22.2i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 22.9iT - 2.19e3T^{2} \)
17 \( 1 + (-32.3 + 55.9i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (15.6 - 9.06i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-76.2 + 44.0i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 308. iT - 2.43e4T^{2} \)
31 \( 1 + (82.5 + 47.6i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (83.8 + 145. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 271.T + 6.89e4T^{2} \)
43 \( 1 - 225.T + 7.95e4T^{2} \)
47 \( 1 + (-31.1 - 54.0i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (19.9 + 11.5i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-41.0 + 71.1i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-424. + 245. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (278. - 482. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 669. iT - 3.57e5T^{2} \)
73 \( 1 + (184. + 106. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-288. - 499. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 1.28e3T + 5.71e5T^{2} \)
89 \( 1 + (-511. - 885. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 472. iT - 9.12e5T^{2} \)
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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

−13.53135305286601004211087086723, −11.56525758143011832818979385797, −10.68195929597427381857856053811, −9.535678393043056129048010843343, −8.362896797626614949047612099983, −7.66619165820513770096458575103, −6.83021943523649006818010234143, −5.58316871105075997578567988949, −2.25749916069693788703317603292, −0.65050968379241714732977848121, 1.82793788144686942908548178008, 3.17581622849684763826752785219, 5.10078704167656333875793357496, 7.59241689522227835781118582648, 8.477151903519409298442883015044, 9.167565582230994422037985777329, 10.33890288932630646338662443445, 10.81000153541666649113180828778, 12.10011166704016808463569920073, 13.05152770118501406788186761253

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