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Properties

Label 2-700-100.87-c1-0-39
Degree $2$
Conductor $700$
Sign $0.955 - 0.295i$
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  + (−1.41 − 0.0483i)2-s + (−0.993 + 1.94i)3-s + (1.99 + 0.136i)4-s + (−0.0255 − 2.23i)5-s + (1.49 − 2.70i)6-s + (0.707 − 0.707i)7-s + (−2.81 − 0.289i)8-s + (−1.04 − 1.44i)9-s + (−0.0719 + 3.16i)10-s + (−0.720 + 0.991i)11-s + (−2.24 + 3.75i)12-s + (−0.598 + 3.77i)13-s + (−1.03 + 0.965i)14-s + (4.38 + 2.17i)15-s + (3.96 + 0.545i)16-s + (0.871 − 0.444i)17-s + ⋯
L(s)  = 1  + (−0.999 − 0.0341i)2-s + (−0.573 + 1.12i)3-s + (0.997 + 0.0683i)4-s + (−0.0114 − 0.999i)5-s + (0.611 − 1.10i)6-s + (0.267 − 0.267i)7-s + (−0.994 − 0.102i)8-s + (−0.349 − 0.481i)9-s + (−0.0227 + 0.999i)10-s + (−0.217 + 0.298i)11-s + (−0.648 + 1.08i)12-s + (−0.165 + 1.04i)13-s + (−0.276 + 0.257i)14-s + (1.13 + 0.560i)15-s + (0.990 + 0.136i)16-s + (0.211 − 0.107i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.295i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 - 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.955 - 0.295i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (687, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ 0.955 - 0.295i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.786095 + 0.118823i\)
\(L(\frac12)\) \(\approx\) \(0.786095 + 0.118823i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.0483i)T \)
5 \( 1 + (0.0255 + 2.23i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 + (0.993 - 1.94i)T + (-1.76 - 2.42i)T^{2} \)
11 \( 1 + (0.720 - 0.991i)T + (-3.39 - 10.4i)T^{2} \)
13 \( 1 + (0.598 - 3.77i)T + (-12.3 - 4.01i)T^{2} \)
17 \( 1 + (-0.871 + 0.444i)T + (9.99 - 13.7i)T^{2} \)
19 \( 1 + (-2.07 + 6.37i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (0.887 + 5.60i)T + (-21.8 + 7.10i)T^{2} \)
29 \( 1 + (-8.54 + 2.77i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (-6.40 - 2.08i)T + (25.0 + 18.2i)T^{2} \)
37 \( 1 + (-7.20 - 1.14i)T + (35.1 + 11.4i)T^{2} \)
41 \( 1 + (2.05 - 1.49i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + (-6.58 - 6.58i)T + 43iT^{2} \)
47 \( 1 + (-1.31 - 0.667i)T + (27.6 + 38.0i)T^{2} \)
53 \( 1 + (-2.94 - 1.50i)T + (31.1 + 42.8i)T^{2} \)
59 \( 1 + (-5.67 + 4.12i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-2.07 - 1.50i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-5.09 - 9.99i)T + (-39.3 + 54.2i)T^{2} \)
71 \( 1 + (3.11 - 1.01i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (3.90 - 0.619i)T + (69.4 - 22.5i)T^{2} \)
79 \( 1 + (2.41 + 7.42i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (0.861 - 0.439i)T + (48.7 - 67.1i)T^{2} \)
89 \( 1 + (4.36 - 6.00i)T + (-27.5 - 84.6i)T^{2} \)
97 \( 1 + (-5.07 + 9.95i)T + (-57.0 - 78.4i)T^{2} \)
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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

−10.19442281359922345593300981010, −9.761495826701159050159854873890, −8.933685529760620147225021875229, −8.190668027600687936443613716922, −7.08541898171418799832646853243, −6.05539725033343159099670155445, −4.78276243781426730676131629729, −4.38150246895367415168330947646, −2.56378057133169710215373747045, −0.860918251304238539272036991343, 0.968271487825205135972905805897, 2.28091973544424253278781852927, 3.38278575332639058219964934130, 5.66092540962432517348592915809, 6.07604593135033332540194676277, 7.11208231584996717528912206313, 7.74248344041173349357407265810, 8.329522089815601379788011342357, 9.799667388366432032824307505541, 10.34509493633177645502698391962

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