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Properties

Label 2-350-35.12-c1-0-10
Degree $2$
Conductor $350$
Sign $-0.0774 + 0.996i$
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  + (0.965 − 0.258i)2-s + (0.788 − 2.94i)3-s + (0.866 − 0.499i)4-s − 3.04i·6-s + (2.01 + 1.71i)7-s + (0.707 − 0.707i)8-s + (−5.43 − 3.13i)9-s + (−1.13 − 1.97i)11-s + (−0.788 − 2.94i)12-s + (3.37 + 3.37i)13-s + (2.38 + 1.13i)14-s + (0.500 − 0.866i)16-s + (−3.34 − 0.896i)17-s + (−6.06 − 1.62i)18-s + (−3.70 + 6.41i)19-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (0.455 − 1.69i)3-s + (0.433 − 0.249i)4-s − 1.24i·6-s + (0.760 + 0.648i)7-s + (0.249 − 0.249i)8-s + (−1.81 − 1.04i)9-s + (−0.342 − 0.594i)11-s + (−0.227 − 0.849i)12-s + (0.936 + 0.936i)13-s + (0.638 + 0.303i)14-s + (0.125 − 0.216i)16-s + (−0.811 − 0.217i)17-s + (−1.42 − 0.382i)18-s + (−0.849 + 1.47i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0774 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0774 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $-0.0774 + 0.996i$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ -0.0774 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55848 - 1.68425i\)
\(L(\frac12)\) \(\approx\) \(1.55848 - 1.68425i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 + 0.258i)T \)
5 \( 1 \)
7 \( 1 + (-2.01 - 1.71i)T \)
good3 \( 1 + (-0.788 + 2.94i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (1.13 + 1.97i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.37 - 3.37i)T + 13iT^{2} \)
17 \( 1 + (3.34 + 0.896i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (3.70 - 6.41i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.776 - 2.89i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 5.27iT - 29T^{2} \)
31 \( 1 + (3 - 1.73i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.19 + 0.588i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 5.19iT - 41T^{2} \)
43 \( 1 + (-0.512 + 0.512i)T - 43iT^{2} \)
47 \( 1 + (-0.123 - 0.459i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-12.3 - 3.31i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-5.19 - 9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.08 + 0.627i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.91 + 10.8i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (0.216 - 0.808i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (8.66 + 5i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.888 - 0.888i)T + 83iT^{2} \)
89 \( 1 + (-4.56 + 7.91i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.18 - 1.18i)T - 97iT^{2} \)
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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

−11.64508940118936977898602131077, −10.78103923298174254980199521550, −8.969812643998103874859884639300, −8.327179836254014432660907274358, −7.39593248436380959457124878960, −6.30921261641611215920998846223, −5.66670024641277242029541663558, −3.95692054521479088791066053406, −2.44614033664507478815000815880, −1.57251975528730482051901351690, 2.61915666940489464675295935097, 3.89776836996580545490280063521, 4.59766108684391111017331855608, 5.39000206003246010211746067121, 6.88177802791229952469459365532, 8.217954702814264721959045874509, 8.858455355024821300340351367327, 10.13682953857009640408292810806, 10.86425099589009954794915571350, 11.26182619187166030965173273465

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