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Properties

Label 2-425-17.2-c1-0-6
Degree $2$
Conductor $425$
Sign $0.914 + 0.403i$
Analytic cond. $3.39364$
Root an. cond. $1.84218$
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  + (−1.66 − 1.66i)2-s + (1.44 − 0.600i)3-s + 3.53i·4-s + (−3.41 − 1.41i)6-s + (−1.37 + 3.30i)7-s + (2.54 − 2.54i)8-s + (−0.379 + 0.379i)9-s + (2.29 + 0.950i)11-s + (2.12 + 5.12i)12-s − 1.25i·13-s + (7.78 − 3.22i)14-s − 1.40·16-s + (3.84 + 1.48i)17-s + 1.26·18-s + (−2.56 − 2.56i)19-s + ⋯
L(s)  = 1  + (−1.17 − 1.17i)2-s + (0.837 − 0.346i)3-s + 1.76i·4-s + (−1.39 − 0.576i)6-s + (−0.518 + 1.25i)7-s + (0.900 − 0.900i)8-s + (−0.126 + 0.126i)9-s + (0.691 + 0.286i)11-s + (0.612 + 1.47i)12-s − 0.349i·13-s + (2.08 − 0.861i)14-s − 0.352·16-s + (0.932 + 0.360i)17-s + 0.297·18-s + (−0.589 − 0.589i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.403i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.914 + 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.914 + 0.403i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (376, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ 0.914 + 0.403i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.893433 - 0.188413i\)
\(L(\frac12)\) \(\approx\) \(0.893433 - 0.188413i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
17 \( 1 + (-3.84 - 1.48i)T \)
good2 \( 1 + (1.66 + 1.66i)T + 2iT^{2} \)
3 \( 1 + (-1.44 + 0.600i)T + (2.12 - 2.12i)T^{2} \)
7 \( 1 + (1.37 - 3.30i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (-2.29 - 0.950i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + 1.25iT - 13T^{2} \)
19 \( 1 + (2.56 + 2.56i)T + 19iT^{2} \)
23 \( 1 + (-8.16 - 3.38i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (-3.35 - 8.09i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (-2.25 + 0.935i)T + (21.9 - 21.9i)T^{2} \)
37 \( 1 + (-4.25 + 1.76i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (1.65 - 3.99i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (5.33 - 5.33i)T - 43iT^{2} \)
47 \( 1 + 11.3iT - 47T^{2} \)
53 \( 1 + (4.02 + 4.02i)T + 53iT^{2} \)
59 \( 1 + (-3.16 + 3.16i)T - 59iT^{2} \)
61 \( 1 + (-0.0929 + 0.224i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + 7.23T + 67T^{2} \)
71 \( 1 + (1.69 - 0.703i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (-2.09 - 5.05i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (8.34 + 3.45i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (-5.17 - 5.17i)T + 83iT^{2} \)
89 \( 1 + 2.19iT - 89T^{2} \)
97 \( 1 + (-3.75 - 9.07i)T + (-68.5 + 68.5i)T^{2} \)
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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

−11.09028217720129063625538255650, −10.03130767833897657852349985210, −9.174416572592750539951638317424, −8.743944133931295307658804620567, −7.964533574597542071046475156162, −6.78724493031497368770361157820, −5.30985595314609937096798349638, −3.31149592752091270799505214140, −2.72655436480540144352652834699, −1.51004209183570363217240745130, 0.866480139129127628502774399354, 3.18761230135222390086610907548, 4.36093875960255563207370292264, 6.05917599549088667774556822948, 6.78595254042098587557565437707, 7.68546200413179985589793985742, 8.501457102004496887606162259710, 9.274965138042943824484448737178, 9.926182599221075967832918669576, 10.67589093254470085444189683948

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