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Properties

Label 2-700-140.19-c1-0-64
Degree $2$
Conductor $700$
Sign $0.808 + 0.587i$
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  + (1.41 + 0.00108i)2-s + (2.85 − 1.64i)3-s + (1.99 + 0.00305i)4-s + (4.03 − 2.32i)6-s + (−1.40 + 2.24i)7-s + (2.82 + 0.00648i)8-s + (3.92 − 6.79i)9-s + (−3.77 + 2.17i)11-s + (5.70 − 3.28i)12-s − 1.67·13-s + (−1.99 + 3.16i)14-s + (3.99 + 0.0122i)16-s + (−1.82 − 3.15i)17-s + (5.55 − 9.60i)18-s + (−1.37 + 2.38i)19-s + ⋯
L(s)  = 1  + (0.999 + 0.000764i)2-s + (1.64 − 0.950i)3-s + (0.999 + 0.00152i)4-s + (1.64 − 0.949i)6-s + (−0.531 + 0.847i)7-s + (0.999 + 0.00229i)8-s + (1.30 − 2.26i)9-s + (−1.13 + 0.657i)11-s + (1.64 − 0.948i)12-s − 0.463·13-s + (−0.532 + 0.846i)14-s + (0.999 + 0.00305i)16-s + (−0.442 − 0.766i)17-s + (1.30 − 2.26i)18-s + (−0.315 + 0.547i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.808 + 0.587i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.808 + 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.808 + 0.587i$
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: \((2,\ 700,\ (\ :1/2),\ 0.808 + 0.587i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.05226 - 1.31715i\)
\(L(\frac12)\) \(\approx\) \(4.05226 - 1.31715i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 - 0.00108i)T \)
5 \( 1 \)
7 \( 1 + (1.40 - 2.24i)T \)
good3 \( 1 + (-2.85 + 1.64i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (3.77 - 2.17i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.67T + 13T^{2} \)
17 \( 1 + (1.82 + 3.15i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.37 - 2.38i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.00 - 1.74i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.77T + 29T^{2} \)
31 \( 1 + (-2.23 - 3.86i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.96 - 2.28i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 1.20iT - 41T^{2} \)
43 \( 1 - 0.424T + 43T^{2} \)
47 \( 1 + (-1.66 - 0.961i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-7.62 + 4.40i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.22 + 10.7i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.41 + 2.54i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.58 - 6.21i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.66iT - 71T^{2} \)
73 \( 1 + (5.17 + 8.96i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.25 + 4.19i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.93iT - 83T^{2} \)
89 \( 1 + (1.55 + 0.896i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 16.3T + 97T^{2} \)
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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

−10.17366628620133629206235413357, −9.446596794914654018495843545515, −8.413480172602284110976160613212, −7.60325646495087702297073674603, −6.99203616611112968912385235338, −5.99243544169789853099040975356, −4.76118610590489422991370977062, −3.44368535848020498434555966767, −2.61930980029491738327788769099, −1.97228560182675699126098306025, 2.28343613117547936765718593220, 3.05129177582266381271238669899, 3.99996802267484648034351673845, 4.60963321088495121276501070913, 5.86686585280636339842098928053, 7.24923283844501811124014080278, 7.84756397220682917068091370801, 8.778530825284340723231896152714, 9.843713513990280963823578984917, 10.52895483471833402977765054078

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