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Properties

Label 2-700-140.19-c1-0-22
Degree $2$
Conductor $700$
Sign $-0.766 - 0.641i$
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  + (0.942 + 1.05i)2-s + (0.780 − 0.450i)3-s + (−0.224 + 1.98i)4-s + (1.21 + 0.398i)6-s + (−1.30 + 2.29i)7-s + (−2.30 + 1.63i)8-s + (−1.09 + 1.89i)9-s + (−3.24 + 1.87i)11-s + (0.720 + 1.65i)12-s + 2.41·13-s + (−3.65 + 0.786i)14-s + (−3.89 − 0.893i)16-s + (−0.291 − 0.505i)17-s + (−3.02 + 0.631i)18-s + (3.07 − 5.33i)19-s + ⋯
L(s)  = 1  + (0.666 + 0.745i)2-s + (0.450 − 0.260i)3-s + (−0.112 + 0.993i)4-s + (0.494 + 0.162i)6-s + (−0.494 + 0.869i)7-s + (−0.815 + 0.578i)8-s + (−0.364 + 0.631i)9-s + (−0.977 + 0.564i)11-s + (0.207 + 0.477i)12-s + 0.671·13-s + (−0.977 + 0.210i)14-s + (−0.974 − 0.223i)16-s + (−0.0707 − 0.122i)17-s + (−0.713 + 0.148i)18-s + (0.706 − 1.22i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.641i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 - 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.766 - 0.641i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ -0.766 - 0.641i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.645020 + 1.77604i\)
\(L(\frac12)\) \(\approx\) \(0.645020 + 1.77604i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.942 - 1.05i)T \)
5 \( 1 \)
7 \( 1 + (1.30 - 2.29i)T \)
good3 \( 1 + (-0.780 + 0.450i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (3.24 - 1.87i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.41T + 13T^{2} \)
17 \( 1 + (0.291 + 0.505i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.07 + 5.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.15 - 3.73i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.435T + 29T^{2} \)
31 \( 1 + (1.26 + 2.19i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-9.78 - 5.65i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 7.35iT - 41T^{2} \)
43 \( 1 - 5.80T + 43T^{2} \)
47 \( 1 + (-10.0 - 5.78i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.69 + 1.55i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.73 - 3.00i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (8.99 + 5.19i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.92 - 8.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.96iT - 71T^{2} \)
73 \( 1 + (4.89 + 8.48i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.397 - 0.229i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.59iT - 83T^{2} \)
89 \( 1 + (-8.55 - 4.94i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 4.54T + 97T^{2} \)
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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

−11.00742552502405181999499881301, −9.597294842094584513928066141490, −8.893000984416756873573925512846, −7.939090691915357056092529555052, −7.42135390972737138308739413364, −6.22728741052887507992355093408, −5.46679950131674812853928460452, −4.57458190294345132732784984115, −3.08514137893833885889689823033, −2.43194641042343938475509745261, 0.74387474228380429959316849461, 2.55363281248899779091612252131, 3.56824271149499195852026352556, 4.12377774663665282082602660288, 5.60845830266553044879902159707, 6.21596246312935162783641235351, 7.50947922184872644835242911025, 8.605341294050091330811866035959, 9.458588040253088576419664165746, 10.36560814441300003003326052323

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