Properties

Label 2-210-15.2-c1-0-9
Degree $2$
Conductor $210$
Sign $-0.190 + 0.981i$
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  + (0.707 − 0.707i)2-s + (−0.931 − 1.46i)3-s − 1.00i·4-s + (2.16 − 0.569i)5-s + (−1.69 − 0.373i)6-s + (0.707 + 0.707i)7-s + (−0.707 − 0.707i)8-s + (−1.26 + 2.72i)9-s + (1.12 − 1.93i)10-s − 6.30i·11-s + (−1.46 + 0.931i)12-s + (−0.977 + 0.977i)13-s + 1.00·14-s + (−2.84 − 2.62i)15-s − 1.00·16-s + (−4.86 + 4.86i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.537 − 0.843i)3-s − 0.500i·4-s + (0.967 − 0.254i)5-s + (−0.690 − 0.152i)6-s + (0.267 + 0.267i)7-s + (−0.250 − 0.250i)8-s + (−0.421 + 0.906i)9-s + (0.356 − 0.610i)10-s − 1.90i·11-s + (−0.421 + 0.268i)12-s + (−0.271 + 0.271i)13-s + 0.267·14-s + (−0.734 − 0.678i)15-s − 0.250·16-s + (−1.18 + 1.18i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.190 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.190 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.190 + 0.981i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ -0.190 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.934039 - 1.13318i\)
\(L(\frac12)\) \(\approx\) \(0.934039 - 1.13318i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (0.931 + 1.46i)T \)
5 \( 1 + (-2.16 + 0.569i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
good11 \( 1 + 6.30iT - 11T^{2} \)
13 \( 1 + (0.977 - 0.977i)T - 13iT^{2} \)
17 \( 1 + (4.86 - 4.86i)T - 17iT^{2} \)
19 \( 1 - 0.285iT - 19T^{2} \)
23 \( 1 + (-4.26 - 4.26i)T + 23iT^{2} \)
29 \( 1 - 3.84T + 29T^{2} \)
31 \( 1 - 6.64T + 31T^{2} \)
37 \( 1 + (-0.317 - 0.317i)T + 37iT^{2} \)
41 \( 1 - 4.55iT - 41T^{2} \)
43 \( 1 + (-2.07 + 2.07i)T - 43iT^{2} \)
47 \( 1 + (6.69 - 6.69i)T - 47iT^{2} \)
53 \( 1 + (-3.12 - 3.12i)T + 53iT^{2} \)
59 \( 1 + 13.0T + 59T^{2} \)
61 \( 1 - 1.09T + 61T^{2} \)
67 \( 1 + (5.63 + 5.63i)T + 67iT^{2} \)
71 \( 1 + 5.42iT - 71T^{2} \)
73 \( 1 + (3.69 - 3.69i)T - 73iT^{2} \)
79 \( 1 - 4.38iT - 79T^{2} \)
83 \( 1 + (1.52 + 1.52i)T + 83iT^{2} \)
89 \( 1 - 8.96T + 89T^{2} \)
97 \( 1 + (-1.50 - 1.50i)T + 97iT^{2} \)
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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

−12.14426284136010829798107251805, −11.19755914443981772702259898634, −10.59644290331386273212066370255, −9.104034498194567954963372168447, −8.206861352831441106593922674313, −6.44470233150105522846585784922, −5.92964108140299805088978615556, −4.80718839586389561963659813570, −2.81389086568044070757467002087, −1.37242599095823206284890245803, 2.61550670295574961636560240702, 4.57346343434114360333979858770, 4.98887735503639322283348798092, 6.46306143082641331420558470941, 7.15732085378312382072242446686, 8.897712998266924308401968178849, 9.818482521613996229322239252646, 10.57042705301724303100850195793, 11.73834532878745229785497547473, 12.68677597736273730523600925385

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