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Properties

Label 2-700-35.27-c1-0-1
Degree $2$
Conductor $700$
Sign $-0.369 - 0.929i$
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  + (−1.83 − 1.83i)3-s + (2.12 + 1.57i)7-s + 3.70i·9-s − 4.70·11-s + (−1.83 − 1.83i)13-s + (0.737 − 0.737i)17-s − 4.75·19-s + (−1 − 6.77i)21-s + (−3.70 + 3.70i)23-s + (1.28 − 1.28i)27-s + 0.701i·29-s + 8.79i·31-s + (8.60 + 8.60i)33-s + (3.70 + 3.70i)37-s + 6.70i·39-s + ⋯
L(s)  = 1  + (−1.05 − 1.05i)3-s + (0.802 + 0.596i)7-s + 1.23i·9-s − 1.41·11-s + (−0.507 − 0.507i)13-s + (0.178 − 0.178i)17-s − 1.09·19-s + (−0.218 − 1.47i)21-s + (−0.771 + 0.771i)23-s + (0.247 − 0.247i)27-s + 0.130i·29-s + 1.58i·31-s + (1.49 + 1.49i)33-s + (0.608 + 0.608i)37-s + 1.07i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.369 - 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.369 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.369 - 0.929i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (657, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ -0.369 - 0.929i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.118171 + 0.174139i\)
\(L(\frac12)\) \(\approx\) \(0.118171 + 0.174139i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.12 - 1.57i)T \)
good3 \( 1 + (1.83 + 1.83i)T + 3iT^{2} \)
11 \( 1 + 4.70T + 11T^{2} \)
13 \( 1 + (1.83 + 1.83i)T + 13iT^{2} \)
17 \( 1 + (-0.737 + 0.737i)T - 17iT^{2} \)
19 \( 1 + 4.75T + 19T^{2} \)
23 \( 1 + (3.70 - 3.70i)T - 23iT^{2} \)
29 \( 1 - 0.701iT - 29T^{2} \)
31 \( 1 - 8.79iT - 31T^{2} \)
37 \( 1 + (-3.70 - 3.70i)T + 37iT^{2} \)
41 \( 1 + 4.75iT - 41T^{2} \)
43 \( 1 + (-5 + 5i)T - 43iT^{2} \)
47 \( 1 + (8.05 - 8.05i)T - 47iT^{2} \)
53 \( 1 + (5 - 5i)T - 53iT^{2} \)
59 \( 1 - 4.75T + 59T^{2} \)
61 \( 1 - 9.50iT - 61T^{2} \)
67 \( 1 + (5 + 5i)T + 67iT^{2} \)
71 \( 1 + 5.40T + 71T^{2} \)
73 \( 1 + (3.11 + 3.11i)T + 73iT^{2} \)
79 \( 1 + 6.70iT - 79T^{2} \)
83 \( 1 + (7.86 + 7.86i)T + 83iT^{2} \)
89 \( 1 + 13.5T + 89T^{2} \)
97 \( 1 + (-5.49 + 5.49i)T - 97iT^{2} \)
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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

−10.83621785752345655937801907929, −10.18539168582786865460466504968, −8.770699446272107582976333219716, −7.85242309430481459065429179833, −7.33502962488371606333684756972, −6.12556448308879151958821965717, −5.46752259478113650830593992957, −4.71050103514066442881232977395, −2.75404319294035051556935970805, −1.60560614553515516178816592482, 0.12106508757464322921694576946, 2.29143367228922813436735895524, 4.08203660962612651457157954382, 4.63391847473389389817894708508, 5.48540227134238584502679189515, 6.40451165155107669183046850475, 7.66520333540118973437936223259, 8.372529455557029231610334897599, 9.809779847274602345886163966357, 10.17717343461318409918226075362

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