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Properties

Label 2-700-140.107-c1-0-46
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: $\chi_{700} (107, \cdot )$
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

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

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