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Properties

Label 2-700-20.3-c1-0-18
Degree $2$
Conductor $700$
Sign $-0.329 - 0.944i$
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  + (0.396 + 1.35i)2-s + (0.945 − 0.945i)3-s + (−1.68 + 1.07i)4-s + (1.65 + 0.909i)6-s + (−0.707 − 0.707i)7-s + (−2.12 − 1.86i)8-s + 1.21i·9-s + 5.00i·11-s + (−0.576 + 2.61i)12-s + (3.27 + 3.27i)13-s + (0.679 − 1.24i)14-s + (1.68 − 3.62i)16-s + (2.38 − 2.38i)17-s + (−1.64 + 0.479i)18-s − 0.428·19-s + ⋯
L(s)  = 1  + (0.280 + 0.959i)2-s + (0.546 − 0.546i)3-s + (−0.842 + 0.537i)4-s + (0.677 + 0.371i)6-s + (−0.267 − 0.267i)7-s + (−0.752 − 0.658i)8-s + 0.403i·9-s + 1.50i·11-s + (−0.166 + 0.754i)12-s + (0.908 + 0.908i)13-s + (0.181 − 0.331i)14-s + (0.421 − 0.906i)16-s + (0.578 − 0.578i)17-s + (−0.387 + 0.113i)18-s − 0.0983·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.329 - 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.329 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.329 - 0.944i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ -0.329 - 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01333 + 1.42759i\)
\(L(\frac12)\) \(\approx\) \(1.01333 + 1.42759i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.396 - 1.35i)T \)
5 \( 1 \)
7 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (-0.945 + 0.945i)T - 3iT^{2} \)
11 \( 1 - 5.00iT - 11T^{2} \)
13 \( 1 + (-3.27 - 3.27i)T + 13iT^{2} \)
17 \( 1 + (-2.38 + 2.38i)T - 17iT^{2} \)
19 \( 1 + 0.428T + 19T^{2} \)
23 \( 1 + (2.54 - 2.54i)T - 23iT^{2} \)
29 \( 1 - 7.51iT - 29T^{2} \)
31 \( 1 - 0.407iT - 31T^{2} \)
37 \( 1 + (3.17 - 3.17i)T - 37iT^{2} \)
41 \( 1 - 5.67T + 41T^{2} \)
43 \( 1 + (-4.01 + 4.01i)T - 43iT^{2} \)
47 \( 1 + (7.13 + 7.13i)T + 47iT^{2} \)
53 \( 1 + (-0.811 - 0.811i)T + 53iT^{2} \)
59 \( 1 + 3.87T + 59T^{2} \)
61 \( 1 - 12.2T + 61T^{2} \)
67 \( 1 + (8.29 + 8.29i)T + 67iT^{2} \)
71 \( 1 + 12.2iT - 71T^{2} \)
73 \( 1 + (3.78 + 3.78i)T + 73iT^{2} \)
79 \( 1 + 0.197T + 79T^{2} \)
83 \( 1 + (-1.86 + 1.86i)T - 83iT^{2} \)
89 \( 1 - 7.05iT - 89T^{2} \)
97 \( 1 + (-5.24 + 5.24i)T - 97iT^{2} \)
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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

−10.52380465050642487808453088150, −9.541901791640311667662879826664, −8.830244937707926302090657569082, −7.85939009444886199395271018036, −7.19259872119535167159833064892, −6.60087927278798850445786244943, −5.32012432406025067801923148738, −4.40338309494763783719202564633, −3.30159178745913285482333615411, −1.76030910922595812460981563490, 0.846992364435731585573953409816, 2.70650353376285078344546976330, 3.47810500839802624629608413661, 4.19243853350190652447691975276, 5.75713372053031661943902510764, 6.10952061144490354058317393919, 8.161534741361536235285731382042, 8.566780116344582278390880513585, 9.488608849101548573453057312084, 10.22923909418731299768947897507

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