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Properties

Label 2-700-7.4-c1-0-10
Degree $2$
Conductor $700$
Sign $0.152 + 0.988i$
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  + (1.28 − 2.22i)3-s + (2.58 − 0.568i)7-s + (−1.79 − 3.11i)9-s + (0.784 − 1.35i)11-s + 5.56·13-s + (−3.58 + 6.20i)17-s + (−1.58 − 2.74i)19-s + (2.05 − 6.47i)21-s + (−2.86 − 4.96i)23-s − 1.53·27-s + 1.96·29-s + (−0.484 + 0.839i)31-s + (−2.01 − 3.49i)33-s + (3.35 + 5.80i)37-s + (7.15 − 12.3i)39-s + ⋯
L(s)  = 1  + (0.741 − 1.28i)3-s + (0.976 − 0.214i)7-s + (−0.599 − 1.03i)9-s + (0.236 − 0.409i)11-s + 1.54·13-s + (−0.869 + 1.50i)17-s + (−0.363 − 0.629i)19-s + (0.448 − 1.41i)21-s + (−0.598 − 1.03i)23-s − 0.296·27-s + 0.365·29-s + (−0.0870 + 0.150i)31-s + (−0.350 − 0.607i)33-s + (0.551 + 0.954i)37-s + (1.14 − 1.98i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.152 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.152 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.152 + 0.988i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (501, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ 0.152 + 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.65007 - 1.41477i\)
\(L(\frac12)\) \(\approx\) \(1.65007 - 1.41477i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 + (-2.58 + 0.568i)T \)
good3 \( 1 + (-1.28 + 2.22i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (-0.784 + 1.35i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 5.56T + 13T^{2} \)
17 \( 1 + (3.58 - 6.20i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.58 + 2.74i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.86 + 4.96i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.96T + 29T^{2} \)
31 \( 1 + (0.484 - 0.839i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.35 - 5.80i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 8.87T + 41T^{2} \)
43 \( 1 + 4.59T + 43T^{2} \)
47 \( 1 + (0.200 + 0.347i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.76 - 8.26i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.28 + 3.95i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.65 + 13.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.431 - 0.746i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.1T + 71T^{2} \)
73 \( 1 + (2 - 3.46i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.43 - 11.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 17.1T + 83T^{2} \)
89 \( 1 + (-2.79 - 4.84i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 0.233T + 97T^{2} \)
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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

−10.50005952352426024234389820930, −8.866662995353297017204193708792, −8.390235010844430797728226977039, −7.938397559692708696535070704455, −6.59738095310243764221972958193, −6.26187848506778767595777986161, −4.63570035320488644889406406178, −3.51994286160821614572495880274, −2.11879179559061837220645262350, −1.24318169359178528094089825795, 1.83992247892992836440496119442, 3.24136711055352975826870897069, 4.18028531927042050893286837400, 4.90601009088188331691221936921, 6.01241869070029754561831047349, 7.35620320890068113084342085647, 8.396123461192877421766056066661, 8.909796836102306188995410510605, 9.696974902118801572416910713908, 10.56393062583902447595799430265

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