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Properties

Label 2-700-28.3-c1-0-20
Degree $2$
Conductor $700$
Sign $0.980 - 0.196i$
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  + (−1.37 − 0.334i)2-s + (0.556 − 0.963i)3-s + (1.77 + 0.919i)4-s + (−1.08 + 1.13i)6-s + (2.32 + 1.26i)7-s + (−2.13 − 1.85i)8-s + (0.880 + 1.52i)9-s + (−1.48 − 0.856i)11-s + (1.87 − 1.20i)12-s + 2.45i·13-s + (−2.77 − 2.51i)14-s + (2.30 + 3.26i)16-s + (5.38 + 3.10i)17-s + (−0.699 − 2.39i)18-s + (0.108 + 0.187i)19-s + ⋯
L(s)  = 1  + (−0.971 − 0.236i)2-s + (0.321 − 0.556i)3-s + (0.888 + 0.459i)4-s + (−0.443 + 0.464i)6-s + (0.878 + 0.477i)7-s + (−0.753 − 0.656i)8-s + (0.293 + 0.508i)9-s + (−0.447 − 0.258i)11-s + (0.541 − 0.346i)12-s + 0.682i·13-s + (−0.740 − 0.672i)14-s + (0.577 + 0.816i)16-s + (1.30 + 0.754i)17-s + (−0.164 − 0.563i)18-s + (0.0248 + 0.0430i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.196i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.980 - 0.196i$
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: \((2,\ 700,\ (\ :1/2),\ 0.980 - 0.196i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19747 + 0.118900i\)
\(L(\frac12)\) \(\approx\) \(1.19747 + 0.118900i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.37 + 0.334i)T \)
5 \( 1 \)
7 \( 1 + (-2.32 - 1.26i)T \)
good3 \( 1 + (-0.556 + 0.963i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (1.48 + 0.856i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.45iT - 13T^{2} \)
17 \( 1 + (-5.38 - 3.10i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.108 - 0.187i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.68 - 3.28i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.47T + 29T^{2} \)
31 \( 1 + (0.0819 - 0.141i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.84 - 6.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.34iT - 41T^{2} \)
43 \( 1 + 1.89iT - 43T^{2} \)
47 \( 1 + (5.85 + 10.1i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.51 + 11.2i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.14 + 3.71i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.06 + 3.50i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.48 - 2.58i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.04iT - 71T^{2} \)
73 \( 1 + (6.59 + 3.80i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-13.8 + 7.97i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.47T + 83T^{2} \)
89 \( 1 + (1.54 - 0.891i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 10.5iT - 97T^{2} \)
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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

−10.29133191569077621708080718438, −9.711871548587242080521328078013, −8.383620175470348688270859309883, −8.141773469636662859659514799852, −7.35839119720958587056653737846, −6.25783434747594788602343040883, −5.15976875856609973765509964886, −3.62510748193804276405241577353, −2.24418774841745207718404980376, −1.47510166339658242695316087543, 0.928283200296125673111796119193, 2.49750166112876960508207664648, 3.81826333670836245822813285126, 5.05990115044188546331080425793, 6.02657376685861646229826443249, 7.36772969455376032605663153334, 7.78331976300229446534430103802, 8.732803172412292727623581671524, 9.643883237181248029363792254060, 10.24148316505262852525374062213

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