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Properties

Label 2-702-117.32-c1-0-3
Degree $2$
Conductor $702$
Sign $-0.693 - 0.720i$
Analytic cond. $5.60549$
Root an. cond. $2.36759$
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  + (0.707 + 0.707i)2-s + 1.00i·4-s + (−0.994 + 0.266i)5-s + (0.339 − 0.0908i)7-s + (−0.707 + 0.707i)8-s + (−0.891 − 0.514i)10-s + (−2.56 + 2.56i)11-s + (0.0241 + 3.60i)13-s + (0.304 + 0.175i)14-s − 1.00·16-s + (1.67 + 2.89i)17-s + (−0.969 − 0.259i)19-s + (−0.266 − 0.994i)20-s − 3.63·22-s + (−0.735 − 1.27i)23-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + 0.500i·4-s + (−0.444 + 0.119i)5-s + (0.128 − 0.0343i)7-s + (−0.250 + 0.250i)8-s + (−0.281 − 0.162i)10-s + (−0.774 + 0.774i)11-s + (0.00670 + 0.999i)13-s + (0.0812 + 0.0469i)14-s − 0.250·16-s + (0.405 + 0.702i)17-s + (−0.222 − 0.0596i)19-s + (−0.0595 − 0.222i)20-s − 0.774·22-s + (−0.153 − 0.265i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.693 - 0.720i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.693 - 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(702\)    =    \(2 \cdot 3^{3} \cdot 13\)
Sign: $-0.693 - 0.720i$
Analytic conductor: \(5.60549\)
Root analytic conductor: \(2.36759\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{702} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 702,\ (\ :1/2),\ -0.693 - 0.720i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.550083 + 1.29197i\)
\(L(\frac12)\) \(\approx\) \(0.550083 + 1.29197i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 \)
13 \( 1 + (-0.0241 - 3.60i)T \)
good5 \( 1 + (0.994 - 0.266i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (-0.339 + 0.0908i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (2.56 - 2.56i)T - 11iT^{2} \)
17 \( 1 + (-1.67 - 2.89i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.969 + 0.259i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.735 + 1.27i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 5.82iT - 29T^{2} \)
31 \( 1 + (-1.48 - 5.54i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-4.96 + 1.33i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-2.41 + 9.02i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-2.37 - 1.36i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (8.18 + 2.19i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + 2.67iT - 53T^{2} \)
59 \( 1 + (-3.42 + 3.42i)T - 59iT^{2} \)
61 \( 1 + (2.86 - 4.95i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-14.5 - 3.90i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-1.53 + 5.74i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-2.47 - 2.47i)T + 73iT^{2} \)
79 \( 1 + (1.76 + 3.05i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.23 + 8.34i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (-2.68 - 10.0i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (3.82 + 14.2i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.84843898079756399739270201201, −9.922229564839770307079762900960, −8.873450220117449889361971703274, −7.973834408315856046369086144820, −7.24191932360371921945051047151, −6.41195163525407427633706835448, −5.27474508613293027130478866482, −4.42030291669683778233017264427, −3.45022186592704238641115405565, −2.01153260413974950765207268877, 0.61340194243034398073179363514, 2.47729486152318254950748743812, 3.42629919444685188936725020402, 4.54500579860120966487678332561, 5.50928101550134717441587476740, 6.29564127562632160993127295818, 7.85740404531376773607359879410, 8.096413781110105985505890745042, 9.556406861920591216224691808974, 10.16619919979784159358848426227

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