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Properties

Label 1-709-709.163-r0-0-0
Degree $1$
Conductor $709$
Sign $-0.979 + 0.199i$
Analytic cond. $3.29258$
Root an. cond. $3.29258$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.530 − 0.847i)2-s + (−0.339 − 0.940i)3-s + (−0.437 + 0.899i)4-s + (0.574 − 0.818i)5-s + (−0.617 + 0.786i)6-s + (−0.697 + 0.716i)7-s + (0.994 − 0.106i)8-s + (−0.769 + 0.638i)9-s + (−0.998 − 0.0532i)10-s + (0.734 − 0.678i)11-s + (0.994 + 0.106i)12-s + (−0.697 − 0.716i)13-s + (0.977 + 0.211i)14-s + (−0.964 − 0.263i)15-s + (−0.617 − 0.786i)16-s + (0.734 + 0.678i)17-s + ⋯
L(s)  = 1  + (−0.530 − 0.847i)2-s + (−0.339 − 0.940i)3-s + (−0.437 + 0.899i)4-s + (0.574 − 0.818i)5-s + (−0.617 + 0.786i)6-s + (−0.697 + 0.716i)7-s + (0.994 − 0.106i)8-s + (−0.769 + 0.638i)9-s + (−0.998 − 0.0532i)10-s + (0.734 − 0.678i)11-s + (0.994 + 0.106i)12-s + (−0.697 − 0.716i)13-s + (0.977 + 0.211i)14-s + (−0.964 − 0.263i)15-s + (−0.617 − 0.786i)16-s + (0.734 + 0.678i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.979 + 0.199i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 709 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.979 + 0.199i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(709\)
Sign: $-0.979 + 0.199i$
Analytic conductor: \(3.29258\)
Root analytic conductor: \(3.29258\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{709} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 709,\ (0:\ ),\ -0.979 + 0.199i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.07677202813 - 0.7604998598i\)
\(L(\frac12)\) \(\approx\) \(-0.07677202813 - 0.7604998598i\)
\(L(1)\) \(\approx\) \(0.4660480349 - 0.5417141861i\)
\(L(1)\) \(\approx\) \(0.4660480349 - 0.5417141861i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad709 \( 1 \)
good2 \( 1 + (-0.530 - 0.847i)T \)
3 \( 1 + (-0.339 - 0.940i)T \)
5 \( 1 + (0.574 - 0.818i)T \)
7 \( 1 + (-0.697 + 0.716i)T \)
11 \( 1 + (0.734 - 0.678i)T \)
13 \( 1 + (-0.697 - 0.716i)T \)
17 \( 1 + (0.734 + 0.678i)T \)
19 \( 1 + (0.994 - 0.106i)T \)
23 \( 1 + (0.185 - 0.982i)T \)
29 \( 1 + (0.949 - 0.314i)T \)
31 \( 1 + (-0.964 - 0.263i)T \)
37 \( 1 + (-0.697 + 0.716i)T \)
41 \( 1 + (0.802 - 0.596i)T \)
43 \( 1 + (-0.887 + 0.461i)T \)
47 \( 1 + (0.185 - 0.982i)T \)
53 \( 1 + (-0.964 - 0.263i)T \)
59 \( 1 + (0.994 - 0.106i)T \)
61 \( 1 + (0.0797 - 0.996i)T \)
67 \( 1 + (0.388 - 0.921i)T \)
71 \( 1 + (-0.998 + 0.0532i)T \)
73 \( 1 + (-0.931 - 0.364i)T \)
79 \( 1 + (-0.132 + 0.991i)T \)
83 \( 1 + (-0.0266 + 0.999i)T \)
89 \( 1 + (-0.237 - 0.971i)T \)
97 \( 1 + (0.484 - 0.874i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.01790542790511289815170110993, −22.357928149509654800428297331734, −21.74505522056926645613082596554, −20.51250236337977667800345187411, −19.68124901229414412141444321529, −18.91128499799596551022257319569, −17.65169455446586586081093882677, −17.45279748110791433670988383737, −16.34997047919653725999623138749, −16.00026797208119195915072013897, −14.71019701122031792809784264558, −14.38917740600830501470607703706, −13.57160722951957656993579311931, −12.04416689125989408368642310027, −11.041061549288888050704258347682, −10.07359676213966585353721159141, −9.66953845819842942279168047078, −9.10902631306705802103696265364, −7.34617993232972237995687052299, −6.9902613789744293804464540015, −5.96927385209711547681406160538, −5.121151648521081214569616146698, −4.07553203638762479497341704811, −3.024081780946762829249684097714, −1.35161830945146320511527744548, 0.50861673664433923289407915212, 1.47672487088319856040353102722, 2.51017489102602702641007419501, 3.40857733378726770874954498894, 4.9975166909210706917535909781, 5.80295352065900086171608275404, 6.80508347152425134080058652079, 8.05508058373485869075569486041, 8.65433379862672172910018702305, 9.55786356810262054615332536094, 10.36228724816022030047593202228, 11.58365190923693562593258504338, 12.29498066902214652961202542226, 12.71490961981493742233523829273, 13.53072155531979672524843956826, 14.42374957354150487764233932445, 16.073344695448121336743616114799, 16.79063269777953408699015577443, 17.372800725781928824739031837615, 18.23032407935332984355217255026, 18.96189707626377904634336517439, 19.66279060384179359307838214205, 20.28965373878012377099701797908, 21.39969153266928309675859254060, 22.164254061854385257819541615159

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