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Properties

Label 2-700-35.9-c1-0-2
Degree $2$
Conductor $700$
Sign $0.389 - 0.921i$
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  + (0.315 − 0.182i)3-s + (1.58 + 2.11i)7-s + (−1.43 + 2.48i)9-s + (−0.682 − 1.18i)11-s + 2.63i·13-s + (1.93 − 1.11i)17-s + (−3.11 + 5.39i)19-s + (0.886 + 0.378i)21-s + (5.71 + 3.29i)23-s + 2.13i·27-s − 5.50·29-s + (−2.25 − 3.89i)31-s + (−0.430 − 0.248i)33-s + (1.81 + 1.04i)37-s + (0.480 + 0.831i)39-s + ⋯
L(s)  = 1  + (0.182 − 0.105i)3-s + (0.600 + 0.799i)7-s + (−0.477 + 0.827i)9-s + (−0.205 − 0.356i)11-s + 0.730i·13-s + (0.468 − 0.270i)17-s + (−0.714 + 1.23i)19-s + (0.193 + 0.0825i)21-s + (1.19 + 0.687i)23-s + 0.411i·27-s − 1.02·29-s + (−0.404 − 0.700i)31-s + (−0.0749 − 0.0432i)33-s + (0.298 + 0.172i)37-s + (0.0769 + 0.133i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.389 - 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.389 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.389 - 0.921i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ 0.389 - 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.25559 + 0.832548i\)
\(L(\frac12)\) \(\approx\) \(1.25559 + 0.832548i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 + (-1.58 - 2.11i)T \)
good3 \( 1 + (-0.315 + 0.182i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (0.682 + 1.18i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.63iT - 13T^{2} \)
17 \( 1 + (-1.93 + 1.11i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.11 - 5.39i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.71 - 3.29i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.50T + 29T^{2} \)
31 \( 1 + (2.25 + 3.89i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.81 - 1.04i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 9.32T + 41T^{2} \)
43 \( 1 - 1.86iT - 43T^{2} \)
47 \( 1 + (-5.94 - 3.43i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-8.78 + 5.06i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.817 + 1.41i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.0197 - 0.0341i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.82 - 3.36i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.27T + 71T^{2} \)
73 \( 1 + (-3.46 + 2i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.66 + 4.61i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 14.7iT - 83T^{2} \)
89 \( 1 + (-0.433 + 0.750i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 16.0iT - 97T^{2} \)
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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

−10.85763761171319669336317185137, −9.591446749588492474551852927842, −8.838508805664879800101704650787, −8.036471836914902829372247757094, −7.33620248472750552830067184213, −5.90744428885837483885372257283, −5.36264493997674071519723360869, −4.15800226004349878561919557246, −2.78300709706525767360071561055, −1.74421861306350091863090892900, 0.803338806595990934575565589383, 2.56858582406735399541777177120, 3.73248501703402517709572996267, 4.72941103393483465338322488805, 5.75556886173723265616166208703, 6.91740038149843779502862972845, 7.62030547510354894956905587827, 8.675330873896492148683512251790, 9.291592749471980613274456251120, 10.52684302250285806451238178425

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