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Properties

Label 2-210-105.23-c1-0-12
Degree $2$
Conductor $210$
Sign $0.278 + 0.960i$
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  + (0.965 − 0.258i)2-s + (−1.71 + 0.255i)3-s + (0.866 − 0.499i)4-s + (−0.962 − 2.01i)5-s + (−1.58 + 0.690i)6-s + (1.50 − 2.17i)7-s + (0.707 − 0.707i)8-s + (2.86 − 0.875i)9-s + (−1.45 − 1.70i)10-s + (0.149 − 0.0864i)11-s + (−1.35 + 1.07i)12-s + (−2.28 − 2.28i)13-s + (0.893 − 2.49i)14-s + (2.16 + 3.21i)15-s + (0.500 − 0.866i)16-s + (0.891 − 3.32i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (−0.989 + 0.147i)3-s + (0.433 − 0.249i)4-s + (−0.430 − 0.902i)5-s + (−0.648 + 0.281i)6-s + (0.569 − 0.821i)7-s + (0.249 − 0.249i)8-s + (0.956 − 0.291i)9-s + (−0.459 − 0.537i)10-s + (0.0451 − 0.0260i)11-s + (−0.391 + 0.311i)12-s + (−0.632 − 0.632i)13-s + (0.238 − 0.665i)14-s + (0.559 + 0.829i)15-s + (0.125 − 0.216i)16-s + (0.216 − 0.807i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.278 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.278 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.278 + 0.960i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ 0.278 + 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01527 - 0.762806i\)
\(L(\frac12)\) \(\approx\) \(1.01527 - 0.762806i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 + 0.258i)T \)
3 \( 1 + (1.71 - 0.255i)T \)
5 \( 1 + (0.962 + 2.01i)T \)
7 \( 1 + (-1.50 + 2.17i)T \)
good11 \( 1 + (-0.149 + 0.0864i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.28 + 2.28i)T + 13iT^{2} \)
17 \( 1 + (-0.891 + 3.32i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-2.38 - 1.37i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.68 - 6.29i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 0.840T + 29T^{2} \)
31 \( 1 + (-4.06 - 7.04i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.58 + 5.93i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 10.1iT - 41T^{2} \)
43 \( 1 + (3.19 + 3.19i)T + 43iT^{2} \)
47 \( 1 + (1.52 - 0.409i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-10.2 - 2.73i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-4.86 - 8.43i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.03 + 10.4i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.24 - 0.332i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 9.00iT - 71T^{2} \)
73 \( 1 + (0.0189 - 0.0707i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-12.0 - 6.94i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.33 - 5.33i)T - 83iT^{2} \)
89 \( 1 + (-7.02 + 12.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.33 - 6.33i)T - 97iT^{2} \)
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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

−12.02669906598274570408582161968, −11.51539666788963502056038739076, −10.47141051729298337411053737162, −9.523662180524985455169740307884, −7.85216183301555679597582254892, −7.02096147796374525015213101120, −5.39025953453797076460127206064, −4.89361471780580661542379833794, −3.68323723887592765939693786986, −1.10639889018738902420311193230, 2.36031207271267461169192562112, 4.14784922913672824397783910127, 5.25614294324846531100562261180, 6.34530393245627614899437534773, 7.15534904898747367315858009582, 8.298289400055599186581376802155, 9.978133744406694454207123615920, 10.98907760422684708783858069439, 11.76885412128600519664406264252, 12.25812270283296372698386523031

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