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Properties

Label 2-700-100.23-c1-0-10
Degree $2$
Conductor $700$
Sign $-0.0456 - 0.998i$
Analytic cond. $5.58952$
Root an. cond. $2.36421$
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  + (−1.41 − 0.0633i)2-s + (0.834 + 1.63i)3-s + (1.99 + 0.179i)4-s + (−1.91 − 1.15i)5-s + (−1.07 − 2.36i)6-s + (0.707 + 0.707i)7-s + (−2.80 − 0.379i)8-s + (−0.221 + 0.304i)9-s + (2.62 + 1.75i)10-s + (0.917 + 1.26i)11-s + (1.36 + 3.41i)12-s + (−0.136 − 0.863i)13-s + (−0.954 − 1.04i)14-s + (0.302 − 4.09i)15-s + (3.93 + 0.713i)16-s + (−1.39 − 0.710i)17-s + ⋯
L(s)  = 1  + (−0.998 − 0.0448i)2-s + (0.481 + 0.945i)3-s + (0.995 + 0.0895i)4-s + (−0.855 − 0.518i)5-s + (−0.438 − 0.965i)6-s + (0.267 + 0.267i)7-s + (−0.990 − 0.134i)8-s + (−0.0737 + 0.101i)9-s + (0.831 + 0.556i)10-s + (0.276 + 0.380i)11-s + (0.395 + 0.984i)12-s + (−0.0379 − 0.239i)13-s + (−0.255 − 0.278i)14-s + (0.0781 − 1.05i)15-s + (0.983 + 0.178i)16-s + (−0.338 − 0.172i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0456 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0456 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.0456 - 0.998i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ -0.0456 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.644792 + 0.674939i\)
\(L(\frac12)\) \(\approx\) \(0.644792 + 0.674939i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.0633i)T \)
5 \( 1 + (1.91 + 1.15i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (-0.834 - 1.63i)T + (-1.76 + 2.42i)T^{2} \)
11 \( 1 + (-0.917 - 1.26i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.136 + 0.863i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (1.39 + 0.710i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (-2.27 - 6.99i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (1.02 - 6.49i)T + (-21.8 - 7.10i)T^{2} \)
29 \( 1 + (0.567 + 0.184i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-6.39 + 2.07i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (5.42 - 0.858i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (-7.10 - 5.16i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (4.06 - 4.06i)T - 43iT^{2} \)
47 \( 1 + (5.50 - 2.80i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (-2.25 + 1.14i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (10.8 + 7.85i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (0.470 - 0.341i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (3.37 - 6.63i)T + (-39.3 - 54.2i)T^{2} \)
71 \( 1 + (6.49 + 2.11i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-13.5 - 2.14i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (-0.112 + 0.346i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-8.12 - 4.13i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + (-6.83 - 9.40i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (6.10 + 11.9i)T + (-57.0 + 78.4i)T^{2} \)
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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

−10.38258303912809974004876334089, −9.633077563237169413180333676390, −9.124518599708033442282803699933, −8.098237023487372340323290559580, −7.71503517000776300245329808940, −6.40779361077536353127176696049, −5.12915859234960088303548688172, −3.98063252591631123035296332139, −3.13901260951189784686449664905, −1.43383289742991894221983444821, 0.69053667398795600460835557324, 2.19083984329514226036375741134, 3.15499935640303010105177060201, 4.64755437568529202791970374324, 6.41059030707678783624520237758, 6.97995022677278772865561851149, 7.63190437238408895957076564216, 8.445335394843962919339039510285, 9.010117863057163119017513756699, 10.38734296658835120647757425491

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