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Properties

Label 1-4025-4025.167-r0-0-0
Degree $1$
Conductor $4025$
Sign $0.279 + 0.960i$
Analytic cond. $18.6920$
Root an. cond. $18.6920$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.226 + 0.974i)2-s + (0.633 + 0.774i)3-s + (−0.897 − 0.441i)4-s + (−0.897 + 0.441i)6-s + (0.633 − 0.774i)8-s + (−0.198 + 0.980i)9-s + (0.0855 − 0.996i)11-s + (−0.226 − 0.974i)12-s + (−0.336 + 0.941i)13-s + (0.610 + 0.791i)16-s + (−0.441 − 0.897i)17-s + (−0.909 − 0.415i)18-s + (−0.985 + 0.170i)19-s + (0.951 + 0.309i)22-s + 24-s + ⋯
L(s)  = 1  + (−0.226 + 0.974i)2-s + (0.633 + 0.774i)3-s + (−0.897 − 0.441i)4-s + (−0.897 + 0.441i)6-s + (0.633 − 0.774i)8-s + (−0.198 + 0.980i)9-s + (0.0855 − 0.996i)11-s + (−0.226 − 0.974i)12-s + (−0.336 + 0.941i)13-s + (0.610 + 0.791i)16-s + (−0.441 − 0.897i)17-s + (−0.909 − 0.415i)18-s + (−0.985 + 0.170i)19-s + (0.951 + 0.309i)22-s + 24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.279 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.279 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4025\)    =    \(5^{2} \cdot 7 \cdot 23\)
Sign: $0.279 + 0.960i$
Analytic conductor: \(18.6920\)
Root analytic conductor: \(18.6920\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4025} (167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4025,\ (0:\ ),\ 0.279 + 0.960i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.179362002 + 0.8848958303i\)
\(L(\frac12)\) \(\approx\) \(1.179362002 + 0.8848958303i\)
\(L(1)\) \(\approx\) \(0.8341277160 + 0.5798756578i\)
\(L(1)\) \(\approx\) \(0.8341277160 + 0.5798756578i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.226 + 0.974i)T \)
3 \( 1 + (0.633 + 0.774i)T \)
11 \( 1 + (0.0855 - 0.996i)T \)
13 \( 1 + (-0.336 + 0.941i)T \)
17 \( 1 + (-0.441 - 0.897i)T \)
19 \( 1 + (-0.985 + 0.170i)T \)
29 \( 1 + (0.985 + 0.170i)T \)
31 \( 1 + (-0.774 - 0.633i)T \)
37 \( 1 + (0.980 + 0.198i)T \)
41 \( 1 + (0.870 - 0.491i)T \)
43 \( 1 + (0.540 - 0.841i)T \)
47 \( 1 + (-0.951 - 0.309i)T \)
53 \( 1 + (0.825 + 0.564i)T \)
59 \( 1 + (0.941 + 0.336i)T \)
61 \( 1 + (0.362 - 0.931i)T \)
67 \( 1 + (0.856 + 0.516i)T \)
71 \( 1 + (-0.921 + 0.389i)T \)
73 \( 1 + (0.884 - 0.466i)T \)
79 \( 1 + (0.0285 - 0.999i)T \)
83 \( 1 + (-0.676 - 0.736i)T \)
89 \( 1 + (-0.998 + 0.0570i)T \)
97 \( 1 + (-0.676 + 0.736i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.25974586529693001583832353131, −17.80203966769090218480759852715, −17.46904878402155287642204953214, −16.52158410784580966008211007656, −15.31866031500924112275470405595, −14.75412820618340039622437611642, −14.21174210373952263107440738682, −13.17840345611547754330400705272, −12.72078939136508785841633875514, −12.49674608062646015912125060024, −11.48294418971468636591361957712, −10.7768397875672952401963291321, −9.957503205153277741555165536551, −9.44202858824022014308289931168, −8.46748472839827970474976015828, −8.1553380793396015983944555105, −7.30025663977338866633846019485, −6.53653796436768722789984254993, −5.549832447408538857001688873082, −4.48630203514981119866066891955, −3.93087319733989013791477968774, −2.904495031009221986688692884395, −2.370820709923609036014720875616, −1.643698934628801423204027165025, −0.757961375722809196465838530821, 0.562179285685591990622744315657, 1.94710556721586848180452486643, 2.82028195031011113210055889327, 3.91948438761730445894217756649, 4.34015694197915938480449555416, 5.17087173574929623067433157150, 5.91851657144398141571447498881, 6.75115262448775588435453412099, 7.46691550136303415113084670400, 8.33456160572796958929407624748, 8.819090898222697894234928177111, 9.38789106385892116995149262028, 10.07723893344025680626173611610, 10.8869276580307349191389654072, 11.53102360973392459206508354739, 12.73754820278829199194978801630, 13.53353445937972317262815460949, 14.04677054305324199414289541028, 14.58714221534114001681551720643, 15.22613331038259004670920199506, 16.049182723334500229119766160388, 16.39624765242488543956735477661, 16.97111993589314343882184806128, 17.808194336945405396099781974951, 18.72687978543551837247154131156

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