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Properties

Label 2-700-35.4-c1-0-10
Degree $2$
Conductor $700$
Sign $-0.952 - 0.305i$
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  + (−2.22 − 1.28i)3-s + (0.568 + 2.58i)7-s + (1.79 + 3.11i)9-s + (0.784 − 1.35i)11-s − 5.56i·13-s + (−6.20 − 3.58i)17-s + (1.58 + 2.74i)19-s + (2.05 − 6.47i)21-s + (−4.96 + 2.86i)23-s − 1.53i·27-s − 1.96·29-s + (−0.484 + 0.839i)31-s + (−3.49 + 2.01i)33-s + (−5.80 + 3.35i)37-s + (−7.15 + 12.3i)39-s + ⋯
L(s)  = 1  + (−1.28 − 0.741i)3-s + (0.214 + 0.976i)7-s + (0.599 + 1.03i)9-s + (0.236 − 0.409i)11-s − 1.54i·13-s + (−1.50 − 0.869i)17-s + (0.363 + 0.629i)19-s + (0.448 − 1.41i)21-s + (−1.03 + 0.598i)23-s − 0.296i·27-s − 0.365·29-s + (−0.0870 + 0.150i)31-s + (−0.607 + 0.350i)33-s + (−0.954 + 0.551i)37-s + (−1.14 + 1.98i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 - 0.305i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.952 - 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.952 - 0.305i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ -0.952 - 0.305i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0180074 + 0.115093i\)
\(L(\frac12)\) \(\approx\) \(0.0180074 + 0.115093i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.568 - 2.58i)T \)
good3 \( 1 + (2.22 + 1.28i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (-0.784 + 1.35i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.56iT - 13T^{2} \)
17 \( 1 + (6.20 + 3.58i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.58 - 2.74i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.96 - 2.86i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.96T + 29T^{2} \)
31 \( 1 + (0.484 - 0.839i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.80 - 3.35i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.87T + 41T^{2} \)
43 \( 1 - 4.59iT - 43T^{2} \)
47 \( 1 + (-0.347 + 0.200i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-8.26 - 4.76i)T + (26.5 + 45.8i)T^{2} \)
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 \( 1 + (0.746 + 0.431i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.1T + 71T^{2} \)
73 \( 1 + (-3.46 - 2i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.43 + 11.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 17.1iT - 83T^{2} \)
89 \( 1 + (2.79 + 4.84i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 0.233iT - 97T^{2} \)
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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

−10.20589298430774858474364760318, −9.052689324290458758750609331530, −8.136586799518527576014158960012, −7.22244522916125985096640718447, −6.18996061077473341497070849683, −5.64366416440386471088934947004, −4.84593718916413228859768950143, −3.14280146251358046014050548756, −1.69722679243895550985657680103, −0.07011486173002554741137250524, 1.85599518192261046685646769275, 4.14438590264126577180142571076, 4.27484795495856312337052704296, 5.43957687205248309930953262295, 6.62980230561231165660315524464, 6.98084918420219707175951617468, 8.484601100115833706762514942510, 9.419004010543291745137456887219, 10.31188592174344276487514604116, 10.86451040106131199264674061027

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