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Properties

Label 2-210-35.27-c1-0-4
Degree $2$
Conductor $210$
Sign $0.774 + 0.632i$
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 + (−0.707 − 0.707i)3-s + 1.00i·4-s + (2.23 − 0.0743i)5-s + 1.00i·6-s + (0.781 + 2.52i)7-s + (0.707 − 0.707i)8-s + 1.00i·9-s + (−1.63 − 1.52i)10-s + 4.90·11-s + (0.707 − 0.707i)12-s + (−3.41 − 3.41i)13-s + (1.23 − 2.33i)14-s + (−1.63 − 1.52i)15-s − 1.00·16-s + (2.74 − 2.74i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.408 − 0.408i)3-s + 0.500i·4-s + (0.999 − 0.0332i)5-s + 0.408i·6-s + (0.295 + 0.955i)7-s + (0.250 − 0.250i)8-s + 0.333i·9-s + (−0.516 − 0.483i)10-s + 1.47·11-s + (0.204 − 0.204i)12-s + (−0.946 − 0.946i)13-s + (0.330 − 0.625i)14-s + (−0.421 − 0.394i)15-s − 0.250·16-s + (0.666 − 0.666i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.774 + 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.774 + 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(210\)    =    \(2 \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.774 + 0.632i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{210} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 210,\ (\ :1/2),\ 0.774 + 0.632i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.971774 - 0.346034i\)
\(L(\frac12)\) \(\approx\) \(0.971774 - 0.346034i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (-2.23 + 0.0743i)T \)
7 \( 1 + (-0.781 - 2.52i)T \)
good11 \( 1 - 4.90T + 11T^{2} \)
13 \( 1 + (3.41 + 3.41i)T + 13iT^{2} \)
17 \( 1 + (-2.74 + 2.74i)T - 17iT^{2} \)
19 \( 1 - 1.26T + 19T^{2} \)
23 \( 1 + (1.05 - 1.05i)T - 23iT^{2} \)
29 \( 1 + 4.76iT - 29T^{2} \)
31 \( 1 - 7.05iT - 31T^{2} \)
37 \( 1 + (4.74 + 4.74i)T + 37iT^{2} \)
41 \( 1 - 5.44iT - 41T^{2} \)
43 \( 1 + (7.58 - 7.58i)T - 43iT^{2} \)
47 \( 1 + (-0.734 + 0.734i)T - 47iT^{2} \)
53 \( 1 + (-1.26 + 1.26i)T - 53iT^{2} \)
59 \( 1 - 1.39T + 59T^{2} \)
61 \( 1 - 2.29iT - 61T^{2} \)
67 \( 1 + (-3.05 - 3.05i)T + 67iT^{2} \)
71 \( 1 + 13.0T + 71T^{2} \)
73 \( 1 + (-5.04 - 5.04i)T + 73iT^{2} \)
79 \( 1 + 14.8iT - 79T^{2} \)
83 \( 1 + (9.29 + 9.29i)T + 83iT^{2} \)
89 \( 1 + 15.3T + 89T^{2} \)
97 \( 1 + (8.42 - 8.42i)T - 97iT^{2} \)
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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

−12.09662482755141149611808104538, −11.51034948626628521470693557601, −10.14949592875143413522780533106, −9.478004791234512168061184430011, −8.489530247512905140068836971443, −7.20494323879070620373096000811, −6.02124929674268618899331823990, −5.01036668166478232092589050913, −2.89754686944664405534522764687, −1.50674180868683001951881449187, 1.54966420676474557161865728307, 4.00963607431349407008361144069, 5.24019269368256042983844855898, 6.46586085443368316146827304782, 7.16138834545618863419912544747, 8.702336114873190320203000622348, 9.672665353448562917415944691695, 10.21546132572550496458987671797, 11.32210036537774110057904274210, 12.33236502869262435710313899917

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