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Properties

Label 2-210-105.2-c1-0-11
Degree $2$
Conductor $210$
Sign $0.805 + 0.592i$
Analytic cond. $1.67685$
Root an. cond. $1.29493$
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  + (−0.258 − 0.965i)2-s + (0.933 + 1.45i)3-s + (−0.866 + 0.499i)4-s + (−0.136 − 2.23i)5-s + (1.16 − 1.27i)6-s + (2.35 − 1.20i)7-s + (0.707 + 0.707i)8-s + (−1.25 + 2.72i)9-s + (−2.12 + 0.709i)10-s + (3.01 − 1.73i)11-s + (−1.53 − 0.796i)12-s + (−0.354 + 0.354i)13-s + (−1.77 − 1.96i)14-s + (3.12 − 2.28i)15-s + (0.500 − 0.866i)16-s + (3.12 + 0.838i)17-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (0.539 + 0.842i)3-s + (−0.433 + 0.249i)4-s + (−0.0611 − 0.998i)5-s + (0.476 − 0.522i)6-s + (0.889 − 0.456i)7-s + (0.249 + 0.249i)8-s + (−0.418 + 0.908i)9-s + (−0.670 + 0.224i)10-s + (0.908 − 0.524i)11-s + (−0.443 − 0.229i)12-s + (−0.0982 + 0.0982i)13-s + (−0.474 − 0.523i)14-s + (0.807 − 0.589i)15-s + (0.125 − 0.216i)16-s + (0.758 + 0.203i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.805 + 0.592i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ 0.805 + 0.592i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27258 - 0.417637i\)
\(L(\frac12)\) \(\approx\) \(1.27258 - 0.417637i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
3 \( 1 + (-0.933 - 1.45i)T \)
5 \( 1 + (0.136 + 2.23i)T \)
7 \( 1 + (-2.35 + 1.20i)T \)
good11 \( 1 + (-3.01 + 1.73i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.354 - 0.354i)T - 13iT^{2} \)
17 \( 1 + (-3.12 - 0.838i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-2.42 - 1.39i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.64 - 1.51i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 1.97T + 29T^{2} \)
31 \( 1 + (4.44 + 7.69i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (8.09 - 2.16i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 7.96iT - 41T^{2} \)
43 \( 1 + (6.03 - 6.03i)T - 43iT^{2} \)
47 \( 1 + (0.617 + 2.30i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.55 - 5.79i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-2.18 - 3.78i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.79 + 6.57i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.64 - 13.6i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 3.96iT - 71T^{2} \)
73 \( 1 + (-3.68 - 0.988i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-7.48 - 4.32i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (10.4 + 10.4i)T + 83iT^{2} \)
89 \( 1 + (-3.59 + 6.23i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-11.6 - 11.6i)T + 97iT^{2} \)
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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

−11.94625425856261654376163448594, −11.39779824476906361166595290406, −10.16379246156275024903710771319, −9.444102654182419171634874463184, −8.436397327244132481046546875637, −7.81409938305391345938653180837, −5.57544276221272113377075303975, −4.45084872482705934252828147335, −3.59347308120710854606321808018, −1.57977096382891851446864622135, 1.90327652277579951249420115220, 3.58756070380168403462700326344, 5.39320842173960214510905575699, 6.65272829825103711671992470525, 7.34202405643441313263530303429, 8.270316388409911677671633994804, 9.238513451343610594976658995084, 10.40908348348529299638246279238, 11.76352421893147934971477118615, 12.33740781979538955421070587458

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