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Properties

Label 2-700-140.123-c1-0-22
Degree $2$
Conductor $700$
Sign $0.993 - 0.111i$
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  + (0.366 − 1.36i)2-s + (0.525 + 1.96i)3-s + (−1.73 − i)4-s + 2.87·6-s + (2.45 + 0.978i)7-s + (−2 + 1.99i)8-s + (−0.976 + 0.563i)9-s + (1.05 − 3.92i)12-s + (2.23 − 2.99i)14-s + (1.99 + 3.46i)16-s + (0.412 + 1.53i)18-s + (−0.627 + 5.33i)21-s + (8.61 + 2.30i)23-s + (−4.97 − 2.87i)24-s + (2.69 + 2.69i)27-s + (−3.27 − 4.15i)28-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)2-s + (0.303 + 1.13i)3-s + (−0.866 − 0.5i)4-s + 1.17·6-s + (0.929 + 0.369i)7-s + (−0.707 + 0.707i)8-s + (−0.325 + 0.187i)9-s + (0.303 − 1.13i)12-s + (0.597 − 0.801i)14-s + (0.499 + 0.866i)16-s + (0.0972 + 0.362i)18-s + (−0.136 + 1.16i)21-s + (1.79 + 0.481i)23-s + (−1.01 − 0.586i)24-s + (0.517 + 0.517i)27-s + (−0.619 − 0.784i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.111i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.993 - 0.111i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (543, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ 0.993 - 0.111i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.91202 + 0.106683i\)
\(L(\frac12)\) \(\approx\) \(1.91202 + 0.106683i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.366 + 1.36i)T \)
5 \( 1 \)
7 \( 1 + (-2.45 - 0.978i)T \)
good3 \( 1 + (-0.525 - 1.96i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 13iT^{2} \)
17 \( 1 + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-8.61 - 2.30i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 10.7iT - 29T^{2} \)
31 \( 1 + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 9.87T + 41T^{2} \)
43 \( 1 + (-2.56 + 2.56i)T - 43iT^{2} \)
47 \( 1 + (-2.56 + 9.56i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.80 + 13.5i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-11.1 + 2.99i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (11.3 - 11.3i)T - 83iT^{2} \)
89 \( 1 + (-16.0 + 9.24i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 97iT^{2} \)
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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

−10.70838355819460242764334104883, −9.661738922716089475254456429982, −8.981160088905628110273420069486, −8.392850963491811035805429343121, −6.94388042852336657505170871125, −5.21931869793364032573804257068, −5.00829314286767242305005458097, −3.81594203878196804645735002164, −2.98223103798158503775370978490, −1.52894681618869438209155325472, 1.07387959994689571930248599899, 2.65774135140634924413377543053, 4.19179926944345570524004576559, 5.06611891363606150915921860008, 6.22905770929840746329325269510, 7.04342338429909228479543116658, 7.71409346118178223890851401159, 8.312059725070778155924623881364, 9.187868143536258631895056561823, 10.38366371152418014009148218858

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