Properties

Label 2-700-20.7-c1-0-52
Degree $2$
Conductor $700$
Sign $-0.998 + 0.0462i$
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  + (1.08 − 0.903i)2-s + (−1.00 − 1.00i)3-s + (0.368 − 1.96i)4-s + (−2.00 − 0.186i)6-s + (−0.707 + 0.707i)7-s + (−1.37 − 2.47i)8-s − 0.967i·9-s + 0.466i·11-s + (−2.35 + 1.60i)12-s + (2.66 − 2.66i)13-s + (−0.131 + 1.40i)14-s + (−3.72 − 1.45i)16-s + (−3.26 − 3.26i)17-s + (−0.874 − 1.05i)18-s − 6.88·19-s + ⋯
L(s)  = 1  + (0.769 − 0.638i)2-s + (−0.581 − 0.581i)3-s + (0.184 − 0.982i)4-s + (−0.819 − 0.0762i)6-s + (−0.267 + 0.267i)7-s + (−0.485 − 0.874i)8-s − 0.322i·9-s + 0.140i·11-s + (−0.679 + 0.464i)12-s + (0.738 − 0.738i)13-s + (−0.0350 + 0.376i)14-s + (−0.931 − 0.362i)16-s + (−0.791 − 0.791i)17-s + (−0.206 − 0.248i)18-s − 1.58·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0462i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0462i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.998 + 0.0462i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ -0.998 + 0.0462i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0325131 - 1.40516i\)
\(L(\frac12)\) \(\approx\) \(0.0325131 - 1.40516i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.08 + 0.903i)T \)
5 \( 1 \)
7 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 + (1.00 + 1.00i)T + 3iT^{2} \)
11 \( 1 - 0.466iT - 11T^{2} \)
13 \( 1 + (-2.66 + 2.66i)T - 13iT^{2} \)
17 \( 1 + (3.26 + 3.26i)T + 17iT^{2} \)
19 \( 1 + 6.88T + 19T^{2} \)
23 \( 1 + (-2.22 - 2.22i)T + 23iT^{2} \)
29 \( 1 + 2.62iT - 29T^{2} \)
31 \( 1 + 3.30iT - 31T^{2} \)
37 \( 1 + (-7.25 - 7.25i)T + 37iT^{2} \)
41 \( 1 + 2.58T + 41T^{2} \)
43 \( 1 + (1.82 + 1.82i)T + 43iT^{2} \)
47 \( 1 + (2.36 - 2.36i)T - 47iT^{2} \)
53 \( 1 + (-7.71 + 7.71i)T - 53iT^{2} \)
59 \( 1 - 11.4T + 59T^{2} \)
61 \( 1 - 2.41T + 61T^{2} \)
67 \( 1 + (-10.8 + 10.8i)T - 67iT^{2} \)
71 \( 1 + 9.62iT - 71T^{2} \)
73 \( 1 + (2.29 - 2.29i)T - 73iT^{2} \)
79 \( 1 + 2.91T + 79T^{2} \)
83 \( 1 + (8.69 + 8.69i)T + 83iT^{2} \)
89 \( 1 - 1.89iT - 89T^{2} \)
97 \( 1 + (-6.04 - 6.04i)T + 97iT^{2} \)
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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

−10.23363132145211702113259284184, −9.385452222881229267654494759355, −8.369797858512951003629706218270, −6.93177327082891966737495363914, −6.33445089464543876683636104056, −5.56305320974856381654830760388, −4.45352989689631014317352056014, −3.33315913176603393728848085973, −2.10870269254651401756054524400, −0.59511215365751618482602305355, 2.30878506738457833217487517376, 3.93024208848350978450778129986, 4.38423737873726287403233339219, 5.50551330229273836295371664110, 6.35277202566303956797229725955, 7.01944599364065309937091395938, 8.341019283427664905325971128731, 8.871173296731294515923364097892, 10.27419781988988470864290219844, 11.00300121415943588339516547160

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