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Properties

Label 2-700-28.27-c1-0-39
Degree $2$
Conductor $700$
Sign $-0.809 + 0.587i$
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  + (−1.37 − 0.331i)2-s − 2.13·3-s + (1.78 + 0.910i)4-s + (2.93 + 0.707i)6-s + (1.19 − 2.35i)7-s + (−2.14 − 1.84i)8-s + 1.56·9-s − 2.33i·11-s + (−3.80 − 1.94i)12-s + 1.09i·13-s + (−2.42 + 2.84i)14-s + (2.34 + 3.24i)16-s + 4.98i·17-s + (−2.14 − 0.516i)18-s + 2.57·19-s + ⋯
L(s)  = 1  + (−0.972 − 0.234i)2-s − 1.23·3-s + (0.890 + 0.455i)4-s + (1.19 + 0.288i)6-s + (0.453 − 0.891i)7-s + (−0.759 − 0.650i)8-s + 0.520·9-s − 0.703i·11-s + (−1.09 − 0.561i)12-s + 0.302i·13-s + (−0.649 + 0.760i)14-s + (0.585 + 0.810i)16-s + 1.20i·17-s + (−0.506 − 0.121i)18-s + 0.590·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.809 + 0.587i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.809 + 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.809 + 0.587i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ -0.809 + 0.587i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.107490 - 0.331159i\)
\(L(\frac12)\) \(\approx\) \(0.107490 - 0.331159i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.37 + 0.331i)T \)
5 \( 1 \)
7 \( 1 + (-1.19 + 2.35i)T \)
good3 \( 1 + 2.13T + 3T^{2} \)
11 \( 1 + 2.33iT - 11T^{2} \)
13 \( 1 - 1.09iT - 13T^{2} \)
17 \( 1 - 4.98iT - 17T^{2} \)
19 \( 1 - 2.57T + 19T^{2} \)
23 \( 1 + 6.04iT - 23T^{2} \)
29 \( 1 + 0.561T + 29T^{2} \)
31 \( 1 + 6.59T + 31T^{2} \)
37 \( 1 - 5.49T + 37T^{2} \)
41 \( 1 + 8.48iT - 41T^{2} \)
43 \( 1 - 1.32iT - 43T^{2} \)
47 \( 1 + 9.74T + 47T^{2} \)
53 \( 1 + 8.58T + 53T^{2} \)
59 \( 1 + 14.3T + 59T^{2} \)
61 \( 1 + 0.620iT - 61T^{2} \)
67 \( 1 - 4.71iT - 67T^{2} \)
71 \( 1 + 11.9iT - 71T^{2} \)
73 \( 1 + 9.96iT - 73T^{2} \)
79 \( 1 + 10.6iT - 79T^{2} \)
83 \( 1 - 3.86T + 83T^{2} \)
89 \( 1 + 2.82iT - 89T^{2} \)
97 \( 1 + 14.9iT - 97T^{2} \)
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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

−10.52291488937517958072920496994, −9.341940475294420702396882420790, −8.357504198516034257681425001258, −7.55770576850110117609462840868, −6.54198234832540860006859115863, −5.91802007025431786189523943477, −4.62382228375135634507187568099, −3.39642492563641245931232026511, −1.61334657327578540262100042597, −0.31211304019048541208088032823, 1.42551319814252716825051515199, 2.86373269719429326903815721370, 4.95822568402514944858396537109, 5.48698629452584455551748222451, 6.38461866479730947800544211511, 7.34779511691237654409658668817, 8.100778634309668959315014640678, 9.369610738562379948221970619118, 9.707787101945686598021652821067, 10.98711709320459937246733057224

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