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Properties

Label 2-700-100.23-c1-0-23
Degree $2$
Conductor $700$
Sign $-0.573 - 0.819i$
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  + (−0.456 + 1.33i)2-s + (0.146 + 0.287i)3-s + (−1.58 − 1.22i)4-s + (0.0469 + 2.23i)5-s + (−0.451 + 0.0650i)6-s + (0.707 + 0.707i)7-s + (2.35 − 1.56i)8-s + (1.70 − 2.34i)9-s + (−3.01 − 0.956i)10-s + (3.70 + 5.10i)11-s + (0.119 − 0.634i)12-s + (−0.622 − 3.92i)13-s + (−1.26 + 0.624i)14-s + (−0.636 + 0.341i)15-s + (1.01 + 3.86i)16-s + (4.82 + 2.46i)17-s + ⋯
L(s)  = 1  + (−0.322 + 0.946i)2-s + (0.0846 + 0.166i)3-s + (−0.791 − 0.610i)4-s + (0.0209 + 0.999i)5-s + (−0.184 + 0.0265i)6-s + (0.267 + 0.267i)7-s + (0.833 − 0.552i)8-s + (0.567 − 0.780i)9-s + (−0.953 − 0.302i)10-s + (1.11 + 1.53i)11-s + (0.0343 − 0.183i)12-s + (−0.172 − 1.08i)13-s + (−0.339 + 0.166i)14-s + (−0.164 + 0.0880i)15-s + (0.254 + 0.967i)16-s + (1.17 + 0.596i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.573 - 0.819i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.573 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(700\)    =    \(2^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.573 - 0.819i$
Analytic conductor: \(5.58952\)
Root analytic conductor: \(2.36421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{700} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 700,\ (\ :1/2),\ -0.573 - 0.819i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.629328 + 1.20837i\)
\(L(\frac12)\) \(\approx\) \(0.629328 + 1.20837i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.456 - 1.33i)T \)
5 \( 1 + (-0.0469 - 2.23i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (-0.146 - 0.287i)T + (-1.76 + 2.42i)T^{2} \)
11 \( 1 + (-3.70 - 5.10i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.622 + 3.92i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (-4.82 - 2.46i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (0.873 + 2.68i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (0.931 - 5.87i)T + (-21.8 - 7.10i)T^{2} \)
29 \( 1 + (0.602 + 0.195i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-0.775 + 0.251i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.0897 - 0.0142i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (1.44 + 1.04i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (1.93 - 1.93i)T - 43iT^{2} \)
47 \( 1 + (6.95 - 3.54i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (6.51 - 3.31i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (-10.7 - 7.81i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-7.45 + 5.41i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (6.67 - 13.1i)T + (-39.3 - 54.2i)T^{2} \)
71 \( 1 + (10.7 + 3.47i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-10.0 - 1.59i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (-5.38 + 16.5i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-4.50 - 2.29i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + (8.10 + 11.1i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (0.216 + 0.424i)T + (-57.0 + 78.4i)T^{2} \)
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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

−10.22787425464554207652799144125, −9.897062577616702512033056729886, −9.127097213547043165337249504331, −7.88520226054923638746469219083, −7.24480863986623292434056514803, −6.50802589234799229516388910469, −5.58836182780749285641690519400, −4.38813031552905731096698083495, −3.40833617601064408076725306611, −1.53902002445728790736203704314, 0.930183418442189136654216092186, 1.91912482145413698035531768907, 3.54948177834431651454287438488, 4.40802502256350495854038188574, 5.32621645324835250841467207037, 6.72140696118172629594403576481, 8.036761010182859422101110881921, 8.442883150197135475411320751112, 9.370400130450633836253425520385, 10.08476064345850631810549586149

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