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Properties

Label 2-350-25.14-c1-0-13
Degree $2$
Conductor $350$
Sign $0.628 - 0.777i$
Analytic cond. $2.79476$
Root an. cond. $1.67175$
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  + (0.587 + 0.809i)2-s + (2.84 + 0.923i)3-s + (−0.309 + 0.951i)4-s + (−0.152 − 2.23i)5-s + (0.923 + 2.84i)6-s − i·7-s + (−0.951 + 0.309i)8-s + (4.79 + 3.48i)9-s + (1.71 − 1.43i)10-s + (−2.02 + 1.46i)11-s + (−1.75 + 2.41i)12-s + (−0.572 + 0.788i)13-s + (0.809 − 0.587i)14-s + (1.62 − 6.48i)15-s + (−0.809 − 0.587i)16-s + (1.56 − 0.509i)17-s + ⋯
L(s)  = 1  + (0.415 + 0.572i)2-s + (1.64 + 0.533i)3-s + (−0.154 + 0.475i)4-s + (−0.0682 − 0.997i)5-s + (0.376 + 1.16i)6-s − 0.377i·7-s + (−0.336 + 0.109i)8-s + (1.59 + 1.16i)9-s + (0.542 − 0.453i)10-s + (−0.609 + 0.442i)11-s + (−0.507 + 0.697i)12-s + (−0.158 + 0.218i)13-s + (0.216 − 0.157i)14-s + (0.419 − 1.67i)15-s + (−0.202 − 0.146i)16-s + (0.380 − 0.123i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.628 - 0.777i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.628 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.628 - 0.777i$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ 0.628 - 0.777i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.27899 + 1.08775i\)
\(L(\frac12)\) \(\approx\) \(2.27899 + 1.08775i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.587 - 0.809i)T \)
5 \( 1 + (0.152 + 2.23i)T \)
7 \( 1 + iT \)
good3 \( 1 + (-2.84 - 0.923i)T + (2.42 + 1.76i)T^{2} \)
11 \( 1 + (2.02 - 1.46i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (0.572 - 0.788i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.56 + 0.509i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.61 - 4.98i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (3.90 + 5.37i)T + (-7.10 + 21.8i)T^{2} \)
29 \( 1 + (-0.766 + 2.35i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (2.67 + 8.24i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (6.73 - 9.27i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (6.96 + 5.05i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 5.97iT - 43T^{2} \)
47 \( 1 + (-3.71 - 1.20i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-2.64 - 0.859i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-8.09 - 5.88i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (1.99 - 1.44i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-9.00 + 2.92i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (0.311 - 0.959i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-8.56 - 11.7i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-0.690 + 2.12i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (3.45 - 1.12i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (14.0 - 10.2i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-1.17 - 0.380i)T + (78.4 + 57.0i)T^{2} \)
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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

−11.98863011434235334309085402758, −10.18773184348613373481775175598, −9.684890472172757438453395105260, −8.499840188572966820258565215539, −8.116610922730353441450576918049, −7.18095511750906198614391596670, −5.49526396683035177361477750348, −4.39421940008948699367763780107, −3.68293965639236660770192640216, −2.17704064235041209518068228085, 1.95801266190515808898824792820, 2.99982515070481123521360261513, 3.56884592536747180474563431316, 5.35655621762122769653329075548, 6.79366690762516458534869529575, 7.62674417002781852462643082602, 8.580062653374886614909563287094, 9.510366012646536519955580088076, 10.39374637914886753825718235701, 11.40777734443100355864365553616

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