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Properties

Label 2-700-100.23-c1-0-33
Degree $2$
Conductor $700$
Sign $0.891 - 0.453i$
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  + (−1.29 − 0.570i)2-s + (0.886 + 1.74i)3-s + (1.34 + 1.47i)4-s + (1.52 − 1.63i)5-s + (−0.155 − 2.75i)6-s + (−0.707 − 0.707i)7-s + (−0.905 − 2.67i)8-s + (−0.478 + 0.658i)9-s + (−2.90 + 1.25i)10-s + (2.71 + 3.73i)11-s + (−1.37 + 3.65i)12-s + (0.637 + 4.02i)13-s + (0.511 + 1.31i)14-s + (4.20 + 1.19i)15-s + (−0.355 + 3.98i)16-s + (0.876 + 0.446i)17-s + ⋯
L(s)  = 1  + (−0.915 − 0.403i)2-s + (0.511 + 1.00i)3-s + (0.674 + 0.737i)4-s + (0.679 − 0.733i)5-s + (−0.0634 − 1.12i)6-s + (−0.267 − 0.267i)7-s + (−0.320 − 0.947i)8-s + (−0.159 + 0.219i)9-s + (−0.917 + 0.396i)10-s + (0.817 + 1.12i)11-s + (−0.395 + 1.05i)12-s + (0.176 + 1.11i)13-s + (0.136 + 0.352i)14-s + (1.08 + 0.307i)15-s + (−0.0889 + 0.996i)16-s + (0.212 + 0.108i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.891 - 0.453i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.891 - 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.891 - 0.453i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ 0.891 - 0.453i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35567 + 0.325188i\)
\(L(\frac12)\) \(\approx\) \(1.35567 + 0.325188i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.29 + 0.570i)T \)
5 \( 1 + (-1.52 + 1.63i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (-0.886 - 1.74i)T + (-1.76 + 2.42i)T^{2} \)
11 \( 1 + (-2.71 - 3.73i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (-0.637 - 4.02i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (-0.876 - 0.446i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (1.78 + 5.49i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (0.419 - 2.64i)T + (-21.8 - 7.10i)T^{2} \)
29 \( 1 + (0.643 + 0.209i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-8.63 + 2.80i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (4.07 - 0.645i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (-8.62 - 6.26i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-3.25 + 3.25i)T - 43iT^{2} \)
47 \( 1 + (9.81 - 5.00i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (-0.701 + 0.357i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (-4.76 - 3.46i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (2.88 - 2.09i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (2.80 - 5.49i)T + (-39.3 - 54.2i)T^{2} \)
71 \( 1 + (-6.12 - 1.99i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-7.50 - 1.18i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (-1.47 + 4.52i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (11.6 + 5.95i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + (3.78 + 5.21i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (-3.47 - 6.82i)T + (-57.0 + 78.4i)T^{2} \)
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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

−10.03452683673602494050635952647, −9.600561566918205931086851937469, −9.155843013893217600862832980788, −8.384206130276360965411731256830, −7.07452858819482361722735328896, −6.32961077384082096976610633613, −4.58509019138301362345774196278, −4.06813505516017233111826177621, −2.63819080726670237550971791252, −1.37804451751031978803734852505, 1.11824005122227900808607868257, 2.32472475384159186915431365375, 3.29385114062293143279596631134, 5.54045840954653598990100368955, 6.28883238580782714075306756904, 6.83758071719993369981449897090, 7.995728789615611892313810977646, 8.378864758969511124536999302883, 9.421595790058695232112619310697, 10.31482334594885217302904991386

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