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Properties

Label 2-210-15.14-c2-0-2
Degree $2$
Conductor $210$
Sign $-0.161 - 0.986i$
Analytic cond. $5.72208$
Root an. cond. $2.39208$
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  + 1.41·2-s + (−2.88 − 0.821i)3-s + 2.00·4-s + (−4.52 + 2.12i)5-s + (−4.08 − 1.16i)6-s − 2.64i·7-s + 2.82·8-s + (7.64 + 4.74i)9-s + (−6.39 + 3.01i)10-s + 18.1i·11-s + (−5.77 − 1.64i)12-s + 22.5i·13-s − 3.74i·14-s + (14.8 − 2.42i)15-s + 4.00·16-s − 0.457·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.961 − 0.273i)3-s + 0.500·4-s + (−0.904 + 0.425i)5-s + (−0.680 − 0.193i)6-s − 0.377i·7-s + 0.353·8-s + (0.849 + 0.526i)9-s + (−0.639 + 0.301i)10-s + 1.65i·11-s + (−0.480 − 0.136i)12-s + 1.73i·13-s − 0.267i·14-s + (0.986 − 0.161i)15-s + 0.250·16-s − 0.0268·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.161 - 0.986i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.161 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.161 - 0.986i$
Analytic conductor: \(5.72208\)
Root analytic conductor: \(2.39208\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1),\ -0.161 - 0.986i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.687738 + 0.809590i\)
\(L(\frac12)\) \(\approx\) \(0.687738 + 0.809590i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41T \)
3 \( 1 + (2.88 + 0.821i)T \)
5 \( 1 + (4.52 - 2.12i)T \)
7 \( 1 + 2.64iT \)
good11 \( 1 - 18.1iT - 121T^{2} \)
13 \( 1 - 22.5iT - 169T^{2} \)
17 \( 1 + 0.457T + 289T^{2} \)
19 \( 1 + 30.1T + 361T^{2} \)
23 \( 1 + 12.3T + 529T^{2} \)
29 \( 1 + 4.70iT - 841T^{2} \)
31 \( 1 - 45.2T + 961T^{2} \)
37 \( 1 - 32.9iT - 1.36e3T^{2} \)
41 \( 1 - 22.9iT - 1.68e3T^{2} \)
43 \( 1 + 20.2iT - 1.84e3T^{2} \)
47 \( 1 + 9.67T + 2.20e3T^{2} \)
53 \( 1 - 5.97T + 2.80e3T^{2} \)
59 \( 1 + 112. iT - 3.48e3T^{2} \)
61 \( 1 + 56.3T + 3.72e3T^{2} \)
67 \( 1 - 67.1iT - 4.48e3T^{2} \)
71 \( 1 + 20.9iT - 5.04e3T^{2} \)
73 \( 1 - 97.7iT - 5.32e3T^{2} \)
79 \( 1 + 41.4T + 6.24e3T^{2} \)
83 \( 1 - 121.T + 6.88e3T^{2} \)
89 \( 1 + 26.8iT - 7.92e3T^{2} \)
97 \( 1 - 151. iT - 9.40e3T^{2} \)
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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

−12.17803564361464903896319018437, −11.73652466137786492977585596662, −10.76624941596667811832402263676, −9.847914115473493575306493672656, −8.024274458388316275848873312568, −6.85316109816884291570658361430, −6.54539889626060218055027621957, −4.59183992241477021789717465499, −4.24297499033257243814920384141, −2.00539133192956337367488274056, 0.52873235373171506438856168794, 3.22764070050728178874387865328, 4.39492583177297239867641993061, 5.55597336284482443852775730962, 6.26341522464514607306533284999, 7.81357930436088558730965532003, 8.711698504554694287233257205199, 10.46059322566017172240079665613, 11.00352543740748505921098458007, 11.98804641557178829922382261561

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