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Properties

Label 2-700-100.23-c1-0-11
Degree $2$
Conductor $700$
Sign $-0.971 + 0.238i$
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  + (−1.22 + 0.713i)2-s + (1.38 + 2.72i)3-s + (0.981 − 1.74i)4-s + (−0.521 + 2.17i)5-s + (−3.63 − 2.33i)6-s + (−0.707 − 0.707i)7-s + (0.0445 + 2.82i)8-s + (−3.72 + 5.12i)9-s + (−0.914 − 3.02i)10-s + (3.29 + 4.52i)11-s + (6.10 + 0.255i)12-s + (−0.466 − 2.94i)13-s + (1.36 + 0.358i)14-s + (−6.64 + 1.59i)15-s + (−2.07 − 3.42i)16-s + (−0.336 − 0.171i)17-s + ⋯
L(s)  = 1  + (−0.863 + 0.504i)2-s + (0.800 + 1.57i)3-s + (0.490 − 0.871i)4-s + (−0.233 + 0.972i)5-s + (−1.48 − 0.952i)6-s + (−0.267 − 0.267i)7-s + (0.0157 + 0.999i)8-s + (−1.24 + 1.70i)9-s + (−0.289 − 0.957i)10-s + (0.992 + 1.36i)11-s + (1.76 + 0.0738i)12-s + (−0.129 − 0.816i)13-s + (0.365 + 0.0959i)14-s + (−1.71 + 0.411i)15-s + (−0.518 − 0.855i)16-s + (−0.0817 − 0.0416i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 + 0.238i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.971 + 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.971 + 0.238i$
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: \((2,\ 700,\ (\ :1/2),\ -0.971 + 0.238i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.134540 - 1.11098i\)
\(L(\frac12)\) \(\approx\) \(0.134540 - 1.11098i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 - 0.713i)T \)
5 \( 1 + (0.521 - 2.17i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (-1.38 - 2.72i)T + (-1.76 + 2.42i)T^{2} \)
11 \( 1 + (-3.29 - 4.52i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.466 + 2.94i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (0.336 + 0.171i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (-1.65 - 5.08i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-0.403 + 2.54i)T + (-21.8 - 7.10i)T^{2} \)
29 \( 1 + (4.85 + 1.57i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-3.80 + 1.23i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (9.90 - 1.56i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (-7.26 - 5.28i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-6.36 + 6.36i)T - 43iT^{2} \)
47 \( 1 + (2.63 - 1.34i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (-4.35 + 2.21i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (2.45 + 1.78i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-11.3 + 8.26i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-2.52 + 4.96i)T + (-39.3 - 54.2i)T^{2} \)
71 \( 1 + (-1.25 - 0.407i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (8.73 + 1.38i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (4.91 - 15.1i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-4.71 - 2.40i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + (2.60 + 3.58i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (-2.29 - 4.50i)T + (-57.0 + 78.4i)T^{2} \)
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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

−10.44930028705901754357578519674, −9.876653272674072564397146731293, −9.498071679973818428228561619436, −8.398747813448018240218552999418, −7.61884825376137503154135910564, −6.75011249172952325669074389740, −5.56834730468554668958973589637, −4.35424836169853415377911682956, −3.45062528192754462102624179798, −2.23893579881930611606004472614, 0.72367967830045385534897636597, 1.67209325752910334167406187904, 2.89289560646676004876174009285, 3.92282953286781855697632229863, 5.84846918382104570064010403909, 6.87189250120062301765679700779, 7.48620759882962523388243476756, 8.601538234767354595161702736500, 8.859374721501791382997055888775, 9.441545607884560500828808506195

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