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Properties

Label 2-700-28.3-c1-0-34
Degree $2$
Conductor $700$
Sign $0.0633 - 0.997i$
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  + (1.36 + 0.366i)2-s + (−0.866 + 1.5i)3-s + (1.73 + i)4-s + (−1.73 + 1.73i)6-s + (1.73 − 2i)7-s + (1.99 + 2i)8-s + (0.866 + 0.5i)11-s + (−3 + 1.73i)12-s + 3.46i·13-s + (3.09 − 2.09i)14-s + (1.99 + 3.46i)16-s + (1.5 + 0.866i)17-s + (−2.59 − 4.5i)19-s + (1.50 + 4.33i)21-s + (0.999 + i)22-s + (−0.866 + 0.5i)23-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)2-s + (−0.499 + 0.866i)3-s + (0.866 + 0.5i)4-s + (−0.707 + 0.707i)6-s + (0.654 − 0.755i)7-s + (0.707 + 0.707i)8-s + (0.261 + 0.150i)11-s + (−0.866 + 0.499i)12-s + 0.960i·13-s + (0.827 − 0.560i)14-s + (0.499 + 0.866i)16-s + (0.363 + 0.210i)17-s + (−0.596 − 1.03i)19-s + (0.327 + 0.944i)21-s + (0.213 + 0.213i)22-s + (−0.180 + 0.104i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0633 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.0633 - 0.997i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ 0.0633 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.86696 + 1.75223i\)
\(L(\frac12)\) \(\approx\) \(1.86696 + 1.75223i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 - 0.366i)T \)
5 \( 1 \)
7 \( 1 + (-1.73 + 2i)T \)
good3 \( 1 + (0.866 - 1.5i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (-0.866 - 0.5i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.46iT - 13T^{2} \)
17 \( 1 + (-1.5 - 0.866i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.59 + 4.5i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.866 - 0.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + (0.866 - 1.5i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.5 - 2.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.46iT - 41T^{2} \)
43 \( 1 + 2iT - 43T^{2} \)
47 \( 1 + (4.33 + 7.5i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.59 - 4.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.5 - 2.59i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.59 + 1.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 14iT - 71T^{2} \)
73 \( 1 + (7.5 + 4.33i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.79 + 4.5i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 13.8T + 83T^{2} \)
89 \( 1 + (-13.5 + 7.79i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 17.3iT - 97T^{2} \)
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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

−10.77629528320173452284900013111, −10.15720929389623464779802560158, −8.933377289196913786259985099469, −7.81719863450051670733408317121, −6.96359588317484526266382831054, −6.07796351920432328499891663420, −4.73398837161031379316204394588, −4.61749105909735277153286067376, −3.51047010091817820626720573861, −1.86892389106486513252926780570, 1.19378165881051310898377605501, 2.35650902733792131294866873783, 3.65440012820743696668623553342, 4.90931777196613126457365628592, 5.84611370216459433362085842509, 6.32418442999754625424321297966, 7.49350590964229571422243359738, 8.207163592800488625927395540619, 9.568131031472067279396068753144, 10.58710606276270928645615554705

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